MECHANICS    DEPT. 


Library 


THEORY  AND  CALCULATION 

OF 

ELECTRIC  CIRCUITS 


THEORY  AND  CALCULATION 

OF 

ELECTRIC  CIRCUITS 


BY 
CHARLES  PROTEUS  STEINMETZ,  A.  M.,  PH.  D. 


FIRST  EDITION 
SEVENTH  IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC. 

NEW  YORK:    370  SEVENTH  AVENUE 

LONDON:    6  &  8  BOUVERIE  ST.,  E.  C.  4 

1917 


(7 

S  ?  z 


Engineering 
Library 


r 


COPYRIGHT,  1917,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY,  INC. 


PRINTED   IN   THE   UNITED   STATES   OF  AMEBICA 


THE  MAPLE  PRESS  -  YORK  PA 


PREFACE 

In  the  twenty  years  since  the  first  edition  of  "Theory  and 
Calculation  of  Alternating  Current  Phenomena"  appeared, 
electrical  engineering  has  risen  from  a  small  beginning  to  the 
world's  greatest  industry;  electricity  has  found  its  field,  as  the 
means  of  universal  energy  transmission,  distribution  and  supply, 
and  our  knowledge  of  electrophysics  and  electrical  engineering 
has  increased  many  fold,  so  that  subjects,  which  twenty  years 
ago  could  be  dismissed  with  a  few  pages  discussion,  now  have  ex- 
panded and  require  an  extensive  knowledge  by  every  electrical 
engineer. 

In  the  following  volume  I  have  discussed  the  most  important 
characteristics  of  the  fundamental  conception  of  electrical  engi- 
neering, such  as  electric  conduction,  magnetism,  wave  shape,  the 
meaning  of  reactance  and  similar  terms,  the  problems  of  stability 
and  instability  of  electric  systems,  etc.,  and  also  have  given  a  more 
extended  application  of  the  method  of  complex  quantities,  which 
the  experience  of  these  twenty  years  has  shown  to  be  the  most 
powerful  tool  in  dealing  with  alternating  current  phenomena. 

In  some  respects,  the  following  work,  and  its  companion 
volume,  "Theory  and  Calculation  of  Electrical  Apparatus," 
may  be  considered  as  continuations,  or  rather  as  parts  of  "The- 
ory and  Calculation  of  Alternating  Current  Phenomena."  With 
the  4th  edition,  which,  appeared  nine  years  ago,  "Alternating 
Current  Phenomena"  had  reached  about  the  largest  practical 
bulk,  and  when  rewriting  it  for  the  5th  edition,  it  became  neces- 
sary to  subdivide  it  into  three  volumes,  to  include  at  least  the 
most  necessary  structural  elements  of  our  knowledge  of  electrical 
engineering.  The  subject  matter  thus  has  been  distributed 
into  three  volumes:  " Alternating  Current  Phenomena,"  "Electric 
Circuits,"  and  "Electrical  Apparatus." 

CHARLES  PROTEUS  STEINMETZ. 

SCHENECTADY, 

January,  1917. 


682S04 


CONTENTS 

PAOB 

PREFACE    v 

SECTION  I 
CHAPTER  I.  ELECTRIC  CONDUCTION.     SOLID  AND  LIQUID  CONDUCTORS 

1.  Resistance — Inductance — Capacity 1 

Metallic  Conductors 

2.  Definition — Range — Constancy — Positive     Temperature    Co- 
efficient—Pure Metals— Alloys 2 

3.  Industrial    Importance    and    Cause — Assumed    Constancy — 
Use  in  Temperature  Measurements 3 

Electrolytic  Conductors 

4.  Definition   by    Chemical    Action — Materials — Range — Nega- 
tive Temperature  Coefficient — Volt-ampere    Characteristic — 
Limitation 4 

5.  Chemical  Action — Faraday's  Law — Energy  Transformation — 
Potential   Difference:  Direction — Constancy — Battery — Elec- 
trolytic Cell — Storage  Battery 6 

6.  Polarization      Cell — Volt-ampere      Characteristic — Diffusion 
Current — Transient  Current 8 

7.  Capacity  of  Polarization  Cell — Efficiency — Application  of  it — 
Aluminum  Cell 9 

Pyroelectric  Conductors 

8.  Definition  by  Dropping  Volt-ampere    Characteristic — Maxi- 
mum and   Minimum  Voltage  Points — Ranges — Limitations.      10 

9.  Proportion   of    Ranges — Materials — Insulators    as    Pyroelec- 
trics — Silicon  and  Magnetite  Characteristics 12 

10.  Use  for  Voltage  Limitation — Effect  of  Transient  Voltage — 
Three  Values  of  Current  for  the  same  Voltage — Stability  and 
Instability  Conditions 14 

11.  Wide  Range  of  Pyroelectric  Conductors — Their  Industrial  Use 

— Cause  of  it — Its  Limitations 18 

12.  Unequal  Current  Distribution  and  Luminous  Streak  Conduc- 
tion— Its  Conditions — Permanent  Increase  of  Resistance  and 
Coherer  Action 18 

13.  Stability  by  Series  Resistance 19 

14.  True  Pyroelectric  Conductors  and  Contact  Resistance  Con- 
ductors   . 20 

Carbon 

15.  Industrial     Importance — Types:     Metallic     Carbon,    Amor- 
phous Carbon,  Anthracite 21 

vii 


viii  CONTENTS 

PAGE 
Insulators 

16.  Definition — Quantitative  Distinction  from  Conductors — Nega- 
tive Temperature  Coefficient — Conduction  at  High  Tempera- 
ture, if  not  Destroyed 23 

17.  Destruction  by  High  Temperature — Leakage  Current — Ap- 
parent Positive  Temperature  Coefficient  by  Moisture  Conduc- 
tion  24 

CHAPTER  II.  ELECTRIC  CONDUCTION.     GAS  AND  VAPOR  CONDUCTORS 

18.  Luminescence — Dropping     Volt-ampere     Characteristic     and 
Instability — Three  Classes:  Spark  Conduction,  Arc  Conduc- 
tion,   Electronic   Conduction — Disruptive   Conduction   ...     28 

19.  Spark,    Streamer,    Corona,     Geissler    Tube    Glow — Discon- 
tinuous  and    Disruptive,  Due  to  Steep  Drop  of  Volt-ampere 
Characteristic — Small    Current    and    High    Voltage — Series 
Capacity — Terminal  Drop  and  Stream   Voltage  of   Geissler 
Tube — Voltage  Gradient  and  Resistivity — Arc  Conduction.     29 

20.  Cathode     Spot — Energy     Required     to     Start — Means     of 
Starting  Arc — Continuous  Conduction 31 

21.  Law  of  Arc  Conduction:  Unidirectional  Conduction — Rectifi- 
cation— Alternating     Arcs — Arc     and     Spark     Voltage     and 
Rectifying  Range 32 

22.  Equations  of  Arc  Conductor — Carbon  Arc 34 

Stability  Curve 

23.  Effect  of  Series  Resistance — Stability  Limit — Stability  Curves 
and  Characteristics  of  Arc 36 

24.  Vacuum  Arcs  and  Their  Characteristics 38 

25.  Voltage  Gradient  and  Resistivity 39 

Electronic  Conduction 

26.  Cold  and   Incandescent   Terminals — Unidirectional   Conduc- 
tion and  Rectification 40 

27.  Total  Volt-ampere  Characteristic  of  Gas  and  Vapor  Conduc- 
tion  40 

Review 

28.  Magnitude  of  Resistivity  of  Different  Types  of  Conductors — 
Relation  of  Streak  Conduction  of  Pyroelectric  and  Puncture 

of  Insulators  . 41 

CHAPTER  III.  MAGNETISM:  RELUCTIVITY 

29.  Frohlich's  and  Kennelly's  Laws. 43 

30.  The  Critical  Points  or  Bends  in  the  Reluctivity  Line  of  Com- 
mercial Materials 44 

31.  Unhomogeneity  of  the  Material  as  Cause  of  the  Bends  in  the 
Reluctivity  Line 47 

32.  Reluctivity  at  Low  Fields,  the  Inward  Bend,  and  the  Rising 
Magnetic  Characteristic  as  part  of  an  Unsymmetrical  Hystere- 
sis Cycle 49 


CONTENTS  ix 

PAGE 

33.  Indefiniteness  of  the  B-H  Relation — The  Alternating  Magnetic 
Characteristic — Instability  and  Creepage 50 

34.  The  Area  of  B-H  Relation — Instability  of  extreme  Values — 
Gradual  Approach  to  the  Stable  Magnetization  Curve.    ...     53 

35.  Production  of  Stable  Values  by  Super-position  of  Alternating 
Field — The  Linear  Reluctivity  Law  of  the  Stable  Magnetic 
Characteristic 54 

CHAPTER  IV.  MAGNETISM:  HYSTERESIS 

36.  Molecular     Magnetic     Friction     and     Hysteresis — Magnetic 
Creepage    .    ; 56 

37.  Area  of  Hysteresis  Cycle  as  Measure  of  Loss 57 

38.  Percentage  Loss  or  Inefficiency  of  Magnetic  Cycle 59 

39.  Hysteresis  Law 60 

40.  Probable  Cause  of  the  Increase  of  Hysteresis  Loss  at  High 
Densities 62 

41.  Hysteresis  at  Low  Magnetic  Densities 64 

42.  Variation  of  77  and  n 66 

43.  The  Slope  of  the  Logarithmic  Curve 68 

44.  Discussion  of  Exponent  n 69 

45.  Unsymmetrical  Hysteresis  Cycles  in  Electrical  Apparatus   .    .  73 

46.  Equations    and    Calculation    of    Unsymmetrical    Hysteresis 
Cycles 74 

CHAPTER  V.  MAGNETISM:  MAGNETIC  CONSTANTS 

47.  The  Ferromagnetic  Metals  and  Their  General  Characteristics  .  77 

48.  Iron,  Its  Alloys,  Mixtures  and  Compounds 79 

49.  Cobalt,  Nickel,  Manganese  and  Chromium 80 

50.  Table  of  Constants  and  Curves  of  Magnetic  Characteristics  .  83 

CHAPTER  VI.  MAGNETISM.     MECHANICAL  FORCES 

51.  Industrial    Importance   of    Mechanical    Forces    in    Magnetic 
Field — Their  Destructive  Effects — General  Equations      ...     89 

52.  The    Constant-current    Electromagnet — Its    Equations    and 
Calculations 93 

53.  The   Alternating-current   Electromagnet — Its  Equations — Its 
Efficiency — Discussion 95 

54.  The    Constant-potential    Alternating-current    Electromagnet 
and  Its  Calculations 98 

55.  ohort-circuit  Stresses  in  Alternating-current  Transformers — 
Calculation    of    Force — Relation    to    Leakage    Reactance — 
Numerical  Instance 99 

56.  Relation  of  Leakage  Reactance  of  Transformer  to  Short-cir- 
cuit Forces — Change  by  Re-arrangement  of  Transformer  Coil 
Groups 102 


x  CONTENTS 

PAGE 

57.  Repulsion    between    Conductor    and    Return    Conductor    of 
Electric  Circuit — Calculations  under  Short-circuit  Conditions 

— Instance 106 

58.  General  Equations  of  Mechanical  Forces  in  Magnetic  Fields — 
Discussion 107 

SECTION  II 
CHAPTER  VII.  SHAPING  OF  WAVES:  GENERAL 

59.  The  General  Advantage  of  the  Sine  Wave Ill 

60.  Effect  of  Field  Flux  Distribution  on  Wave  Shape — Odd  and 
Even  Harmonics 114 

61.  Reduction    and    Elimination    of    Harmonics    by    Distributed 
Winding 116 

62.  Elimination  of  Harmonics  by  Fractional  Pitch,  etc 119 

63.  Relative  Objection  of  Harmonics,  and  Specifications  of  Sine 
Wave  by  Current  in  Condenser  Resistance 120 

64.  Some  Typical  Cases  requiring  Wave  Shape  Distortion      .    .    .  123 

CHAPTER  VIII.  SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION 

65.  Current  Waves  in  Saturated  Closed  Magnetic  Circuit,  with 
Sine  Wave  of  Impressed  Voltage 125 

66.  Voltage    Waves    of    a    Saturated    Closed    Magnetic    Circuit 
Traversed  by  a  Sine  Wave  of  Current,  and  their  Excessive 
Peaks 129 

67.  Different  Values  of  Reactance  of  Closed  Magnetic  Circuit,  on 
Constant  Potential,  Constant  Current  and  Peak  Values   .    .    .    132 

68.  Calculation  of  Peak  Value  and  Form  Factor  of   Distorted 
Wave  in  Closed  Magnetic  Circuit 136 

69.  Calculation  of  the  Coefficients  of  the  Peaked  Voltage  Wave  of 
the  Closed  Magnetic  Circuit  Reactance 139 

70.  Calculation  of  Numerical  Values  of  the  Fourier  Series  of  the 
Peaked  Voltage  Wave  of  a  Closed  Magnetic  Circuit  Reactor  .   141 

71.  Reduction  of  Voltage  Peaks  in  Saturated  Magnetic  Circuit, 

by  Limited  Supply  Voltage 143 

72.  Effect  of  Air  Gap  in  Reducing  Saturation  Peak  of  Voltage  in 
Closed  Magnetic  Circuit 145 

73.  Magnetic  Circuit  with  Bridged  or  Partial  Air  Gap 147 

74.  Calculation  of  the  Voltage  Peak  of  the  Bridged  Gap,  and  Its 
Reduction  by  a  Small  Unbridged  Gap 149 

75.  Possible  Danger  and  Industrial  Use  of  High  Voltage  Peaks. 
Their  Limited  Power  Characteristics 151 

CHAPTER  IX.  WAVE  SCREENS.     EVEN  HARMONICS 

76.  Reduction  of  Wave  Distortion  by  "Wave  Screens" — React- 
ance as  Wave  Screen 153 


CONTENTS  xi 

PAGE 

77.  T-connection  or  Resonating  Circuit  as  Wave  Screen — Numer- 
ical Instances 154 

78.  Wave  Screen  Separating  (or  Combining)  Direct  Current  and 
Alternating     Current — Wave     Screen     Separating     Complex 
Alternating  Wave  into  its  Harmonics 156 

79.  Production  of  Even  Harmonics  in  Closed  Magnetic  Circuit   .    .    157 

80.  Conclusions 160 

CHAPTER  X.  INSTABILITY  OF  CIRCUITS:  THE  ARC 

A.  General 

81.  The  Three  Main  Types  of  Instability  of  Electric  Circuits  .    .    165 

82.  Transients 165 

83.  Unstable  Electric  Equilibrium — The  General  Conditions  of 
Instability  of  a  System — The  Three  Different  Forms  of  Insta- 
bility of  Electric  Circuits 162 

84.  Circuit  Elements  Tending  to  Produce  Instability — The  Arc, 
Induction  and  Synchronous  Motors 164 

85.  Permanent  Instability — Condition  of  its  Existence — Cumula- 
tive Oscillations  and  Sustained  Oscillations 165 

B.  The  Arc  as  Unstable  Conductor. 

86.  Dropping  Volt-ampere  Characteristic  of  Arc  and  Its  Equation 
— Series   Resistance  and   Conditions   of   Stability — Stability 
Characteristic  and  Its  Equation 167 

87.  Conditions  of  Stability  of  a  Circuit,  and  Stability  Coefficient  .    169 

88.  Stability    Conditions   of   Arc   on    Constant    Voltage   Supply 
through  Series  Resistance 171 

89.  Stability  Conditions  of  Arc  on  Constant  Current  Supply  with 
Shunted  Resistance 172 

90.  Parallel    Operation    of    Arcs — Conditions    of    Stability    with 
Series  Resistance 175 

91.  Investigation  of  the  Effect  of  Shunted  Capacity  on  a  Circuit 
Traversed  by  Continuous  Current 178 

92.  Capacity  in  Shunt  to  an  Arc,  Affecting  Stability — Resistance 

in  Series  to  Capacity 180 

93.  Investigation  of  the  Stability  Conditions  of  an  Arc  Shunted 

by  Capacity : 181 

94.  Continued  Calculations  and  Investigation  of  Stability  Limit.  .  183 

95.  Capacity,    Inductance  and  Resistance  in  Shunt  to    Direct- 
current  Circuit 186 

96.  Production   of   Oscillations   by    Capacity,    Inductance    and 
Resistance    Shunting  Direct-current  Arc — Arc  as  Generator 
of     Alternating-current     Power — Cumulative     Oscillations — 
Singing  Arc — Rasping  Arc 187 

97.  Instance — Limiting  Resistance  of  Arc  Oscillations 189 

98.  Transient   Arc    Characteristics — Condition    of    Oscillation — 
Limitation  of  Amplitude  of  Oscillation 191 

99.  Calculation  of  Transient  Arc  Characteristic — Instance.  .    194 


xii  CONTENTS 

PAGE 

100.  Instance  of  Stability  of  Transmission  System  due  to  Arcing 
Ground — Continuous  Series  of  Successive  Discharges.    .    .    .    198 

101.  Cumulative  Oscillations  in  High-potential  Transformers    .    .    199 

CHAPTER  XI.  INSTABILITY  OF  CIRCUITS:  INDUCTION  AND  SYNCHRONOUS 
MOTORS 

C.  Instability  of  Induction  Motors 

102.  Instability  of  Electric  Circuits  by  Non-electrical  Causes — 
Instability    Caused   by    Speed-torque   Curve   of    Motor  in 
Relation  to  Load — Instances 201 

103.  Stability  Conditions  of  Induction  Motor  on  Constant  Torque 
Load — Overload  Conditions 204 

104.  Instability  of  Induction   Motor  as  Function  of  the  Speed 
Characteristic  of  the  Load — Load   Requiring  Torque   Pro- 
portional to  Speed 205 

105.  Load  Requiring  Torque  Proportional  to  Square  of  Speed — 
Fan  and  Propeller 207 

D.  Hunting  of  Synchronous  Machines 

106.  Oscillatory  Instability  Typical  of  Synchronous  Machines — 
Oscillatory    Readjustment    of    Synchronous    Machine    with 
Changes  of  Loads 208 

107.  Investigation  of  the  Oscillation  of  Synchronous  Machines — 
Causes  of  the  Damping — Cumulative  Effect  Due  to  Lag  of 
Synchronizing  Force  Behind  Position 210 

108.  Mathematical  Calculations  of  Synchronizing  Power  and  of 
Conditions  of  Instability  of  Synchronous  Machine 213 

CHAPTER  XII.  REACTANCE  OF  INDUCTION  APPARATUS 

109.  Inductance  as  Constant  of  Every  Electric  Circuit — Merging 
of  Magnetic  Field  of  Inductance  with  other  Magnetic  Fields 
and  Its  Industrial  Importance  Regarding  Losses,  M.m.fs.,  etc.  216 

Leakage  Flux  of  Alternating-current  Transformer 

110.  Mutual  Magnetic  Flux  and  Leakage  or  Reactance  Flux  of 
Transformer — Relation  of  Their  Reluctances 217 

111.  Vector    Diagram    of    Transformer    Including    Mutual    and 
Leakage  Fluxes — Combination  of  These  Fluxes 219 

112.  The  Component  Magnetic  Fluxes  of  the  Transformer  and 
Their  Resultant  Fluxes — Magnetic  Distribution  in  Trans- 
former at  Different  Points  of  the  Wave 221 

113.  Symbolic    Representation    of    Relation    between    Magnetic 
Fluxes  and  Voltages  in  Transformer 222 

114.  Arbitrary  Division  of  Transformer  Reactance  into  Primary 
and  Secondary — Subdivision  of  Reactances  by  Assumption 

of  Core  Loss  being  Given  by  Mutual  Flux 223 

115.  Assumption  of  Equality  of  Primary  and  Secondary  Leakage 


CONTENTS  xiii 

PAGE 

Flux — Cases  of  Inequality  of  Primary  and  Secondary  React- 
ance— Division  of  Total  Reactance  in  Proportion  of  Leakage 
Fluxes 224 

116.  Subdivision  of  Reactance  by  Test — Impedance  Test  and  Its 
Meaning — Primary    and    Secondary    Impedance    Test    and 
Subdivision  of  Total  Reactance  by  It 226 

Magnetic  Circuits  of  Induction  Motor 

117.  Mutual  Flux  and  Resultant  Secondary  Flux — True  Induced 
Voltage  and  Resistance  Drop — Magnetic  Fluxes  and  Voltages 

of  Induction  Motor 228 

118.  Application  of  Method  of  True  Induced  Voltage,   and  Re- 
sultant Magnetic  Fluxes,  to  Symbolic  Calculation  of  Poly- 
phase Induction  Motor 230 

CHAPTER  XIII.  REACTANCE  OF  SYNCHRONOUS  MACHINES 

119.  Armature    Reactance — Field    Flux,     Armature    Flux    and 
Resultant  Flux — Its  Effects:  Demagnetization  and  Distor- 
tion, in  Different  Relative  Positions — Corresponding  M.m.f 
Combinations:  M.m.f.    of    Field    and     Counter-m.m.f.     of 
Armature — Effect  on  Resultant  and  on  Leakage  Flux  .    .    .   232 

120.  Corresponding  Theories:  That  of  Synchronous  Reactance  and 
that  of  Armature  Reaction — Discussion  of  Advantages  and  of 
Limitation    of   Synchronous    Reactance   and    of    Armature 
Reaction  Conception 236 

121.  True  Self-inductive  Flux  of  Armature,  and  Mutual  Inductive 
Flux  with  Field  Circuit — Constancy  of   Mutual  Inductive 
Flux  in  Polyphase  Machine  in  Stationary  Condition  of  Load — 
Effect  of  Mutual  Flux  on  Field  Circuit  in  Transient  Condition 
of  Load — Over-shooting  of  Current  at  Sudden  Change,  and 
Momentary  Short-circuit  Current 237 

122.  Subdivision  of  Armature  Reactance  in  Self-inductive    and 
Mutual    Inductive    Reactance    Necessary    in    Transients, 
Representing  Instantaneous  and  Gradual  Effects — Numerical 
Proportions — Squirrel  Cage 238 

123.  Transient  Reactance — Effect  of  Constants  of  Field  Circuit 
on    Armature    Circuit   during  Transient — Transient  React- 
ance in  Hunting  of  Synchronous  Machines 239 

124.  Double  Frequency  Pulsation  of  Field  in  Single-phase  Machine, 
or  Polyphase   Machine  on   Unbalanced  Load — Third  Har- 
monic Voltage  Produced  by  Mutual  Reactance 240 

125.  Calculation  of  Phase  Voltage  and  Terminal  Voltage  Waves  of 
Three-phase   Machine  at   Balanced  Load — Cancellation  of 
Third  Harmonics 241 

126.  Calculation  of  Phase  Voltage  and  Terminal  Voltage  Waves  of 
Three-phase  Machine  at  Unbalanced  Load — Appearance  of 
Third  Harmonics  in  Opposition  to  Each  Other  in  Loaded  and 
Unloaded  Phases — Equal  to  Fundamental  at  Short  Circuit  243 


xiv  CONTENTS 

PAGE 
SECTION  III 

CHAPTER  XIV.  CONSTANT   POTENTIAL    CONSTANT   CURRENT   TRANS- 
FORMATION 

127.  Constant  Current  in  Arc  Lighting— Tendency  to  Constant 
Current  in  Line  Regulation 245 

128.  Constant   Current  by  Inductive   Reactance,    Non-inductive 
Receiver  Circuit 245 

129.  Constant     Current     by     Inductive     Reactance,     Inductive 
Receiver  Circuit 248 

130.  Constant  Current  by  Variable  Inductive  Reactance   ....    250 

131.  Constant  Current  by  Series  Capacity,  with  Inductive  Cir- 
cuit     253 

132.  Constant  Current  by  Resonance 255 

133.  T-Connection 258 

134.  Monocyclic  Square 259 

135.  T-Connection  or  Resonating  Circuit:  General  Equation    .    .    261 

136.  Example 264 

137.  Apparatus  Economy  of  the  Device 265 

138.  Energy  Losses  in  the  Reactances 268 

139.  Example 270 

140.  Effect  of  Variation  of  Frequency 271 

141.  Monocyclic  Square:  General  Equations 273 

142.  Power  and  Apparatus  Economy 275 

143.  Example .   276 

144.  Power  Losses  in  Reactances 277 

145.  Example 279 

146.  General  Discussion:  Character  of  Transformation  by  Power 
Storage  in  Reactances 280 

147.  Relation  of  Power  Storage  to  Apparatus  Economy  of  Dif- 
ferent Combinations 281 

148.  Insertion  of  Polyphase  e.m.fs.   and  Increase   of  Apparatus 
Economy 283 

149.  Problems  and  Systems  for  Investigation 286 

150.  Some  Further  Problems 287 

151.  Effect  of  Distortion  of  Impressed  Voltage  Wave      290 

152.  Distorted  Voltage  on  T-Connections 290 

153.  Distorted  Voltage  on  Monocyclic  Square 293 

154.  General  Conclusions  and  Problems 295 

CHAPTER  XV.  CONSTANT  POTENTIAL  SERIES  OPERATION 

155.  Condition  of  Series  Operation.     Reactor  as  Shunt  Protective 
Device.     Street  Lighting 297 

156.  Constant  Reactance  of  Shunted  Reactor,  and  Its  Limitations  299 

157.  Regulation  by  Saturation  of  Shunted  Reactor 301 

158.  Discussion    .  .   303 


CONTENTS  xv 

PAGE 

159.  Calculation  of  Instance 305 

160.  Approximation  of  Effect  of  Line  Impedance  and  Leakage 
Reactance — Instance 306 

161.  Calculation    of    Effect    of    Line     Impedance    and    Leakage 
Reactance 308 

162.  Effect  of  Wave  Shape  Distortion  by  Saturation  of  Reactor, 

on  Regulation — Instance 310 

CHAPTER  XVI.  LOAD  BALANCE  OF  POLYPHASE  SYSTEMS 

163.  Continuous  and  Alternating  Component  of  Flow  of  Power — 
Effect  of  Alternating  Component  on  Regulation  and  Effi- 
ciency— Balance  by  Energy  Storing  Devices 314 

164.  Power  Equation  of  Single-phase  Circuit 315 

165.  Power  Equation  of  Polyphase  Circuit 316 

166.  Balance  of  Circuit  by  Reactor  in  Circuit  of  Compensating 
Voltage 318 

167.  Balance  by  Capacity  in  Compensating  Circuit 319 

168.  Instance  of  Quarterphase  System — General   Equations  and 
Non-inductive  Load 321 

169.  Quarterphase  System:  Phase  of  Compensating  Voltage  at 
Inductive  Load,  and  Power  Factor  of  System 322 

170.  Quarterphase     System:  Two     Compensating     Voltages     of 
Fixed  Phase  Angle 324 

171.  Balance  of  Three-phase  System — Coefficient  of  Unbalancing 

at  Constant  Phase  Angle  of  Compensating  Voltage     ....   326 

CHAPTER  XVII.  CIRCUITS  WITH  DISTRIBUTED  LEAKAGE 

172.  Industrial  Existence  of  Conductors  with  Distributed  Leakage: 
Leaky  Main  Conductors — Currents  Induced  in  Lead  Armors 

— Conductors  Traversed  by  Stray  Railway  Currents  ....   330 

173.  General  Equations  of  Direct  Current  in  Leaky  Conductor     .   331 

174.  Infinitely  Long  Leaky  Conductor  and  Its  Equivalent  Resist- 
ance— Open   Circuited  Leaky   Conductor — Grounded    Con- 
ductor— Leaky  Conductor  Closed  by  Resistance 332 

175.  Attenuation  Constant  of  Leaky  Conductor — Outflowing  and 
Return  Current — Reflection  at  End  of  Leaky  Conductor  .    .   333 

176.  Instance  of  Protective  Ground  Wire  of  Transmission  Lines  .   335 

177.  Leaky   Alternating-current   Conductor — General   Equations 
of    Current   in   Leaky    Conductor    Having   Impressed    and 
Induced  Alternating  Voltage 336 

178.  Equations  of  Leakage  Current  in  Conductor  Due  to  Induced 
Alternating  Voltage:  Lead  Armor  of  Single  Conductor  Al- 
ternating-current Cable — Special  Cases 337 

179.  Instance  of  Grounded  Lead  Armor  of  Alternating-current 
Cable 339 

180.  Grounded   Conductor   Carrying  Railway  Stray  Currents — 
Instance  .  .341 


xvi  CONTENTS 

CHAPTER  XVIII.  OSCILLATING  CURRENTS 

PAGE 

181.  Introduction 343 

182.  General  Equations 344 

183.  Polar  Cbdrdinates 345 

184.  Loxodromic  Spiral 346 

185.  Impedance  and  Admittance 347 

186.  Inductance 347 

187.  Capacity 348 

188.  Impedance 348 

189.  Admittance      349 

190.  Conductance  and  Susceptance 350 

191.  Circuits  of  Zero  Impedance 351 

192.  Continued 351 

193.  Origin  of  Oscillating  Currents 352 

194.  Oscillating  Discharge 353 

INDEX    .                                                                                                            .  355 


THEORY  AND  CALCULATION  OF 
ELECTRIC  CIRCUITS 


SECTION  I 

CHAPTER  I 

ELECTRIC  CONDUCTION.     SOLID  AND  LIQUID 
CONDUCTORS 

1.  When  electric  power  flows  through  a  circuit,  we  find  phe- 
nomena taking  place  outside  of  the  conductor  which  directs  the 
flow  of  power,  and  also  inside  thereof.     The  phenomena  outside 
of  the  conductor  are  conditions  of  stress  in  space  which  are  called 
the  electric  field,  the  two  main  components  of  the  electric  field 
being  the  electromagnetic  component,  characterized  by  the  cir- 
cuit constant  inductance,  L,  and  the  electrostatic  component, 
characterized  by  the  electric  circuit  constant  capacity,  C.     Inside 
of  the  conductor  we  find  a  conversion  of  energy  into  heat;  that  is, 
electric  power  is  consumed  in  the  conductor  by  what  may  be 
considered  as  a  kind  of  resistance  of  the  conductor  to  the  flow  of 
electric  power,  and  so  we  speak  of  resistance  of  the  conductor  as 
an  electric  quantity,  representing  the  power  consumption  in  the 
conductor. 

Electric  conductors  have  been  classified  and  divided  into  dis- 
tinct groups.  We  must  realize,  however,  that  there  are  no  dis- 
tinct classes  in  nature,  but  a  gradual  transition  from  type  to  type. 

Metallic  Conductors 

2.  The  first  class  of  conductors  are  the  metallic  conductors. 
They  can  best  be  characterized  by  a  negative  statement — that  is, 
metallic  conductors  are  those  conductors  in  which  the  conduction 
of  the  electric  current  converts  energy  into  no  other  form  but  heat. 
That  is,  a  consumption  of  power  takes  place  in  the  metallic  con- 

1 


ELECTRIC  CIRCUITS 


ductors  by  06n version  into  heat,  and  into  heat  only.  Indirectly, 
we  may  get  light,  if  the  heat  produced  raises  the  temperature 
high  enough  to  get  visible  radiation  as  in  the  incandescent  lamp 
filament,  but  this  radiation  is  produced  from  heat,  and  directly 
the  conversion  of  electric  energy  takes  place  into  heat.  Most 
of  the  metallic  conductors  cover,  as  regards  their  specific  resist- 
ance, a  rather  narrow  range,  between  about  1.6  microhm-cm. 
(1.6  X  10~6)  for  copper,  to  about  100  microhm-cm,  for  cast  iron, 
mercury,  high-resistance  alloys,  etc.  They,  therefore,  cover  a 
range  of  less  than  1  to  100. 


RESISTANCE  -TEMPERATURE    CHARACTERISTIC 

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I     PURE  METALS 
II     ALLOYS 

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FIG.  1. 

A  characteristic  of  metallic  conductors  is  that  the  resistance 
is  approximately  constant,  varying  only  slightly  with  the  tem- 
perature, and  this  variation  is  a  rise  of  resistance  with  increase 
of  temperature — that  is,  they  have  a  positive  temperature  co- 
efficient. In  the  pure  metals,  the  resistance  apparently  is  ap- 
proximately proportional  to  the  absolute  temperature — that  is, 
the  temperature  coefficient  of  resistance  is  constant  and  such  that 
the  resistance  plotted  as  function  of  the  temperature  is  a  straight 
line  which  points  toward  the  absolute  zero  of  temperature,  or, 
in  other  words,  which,  prolonged  backward  toward  falling  tern- 


ELECTRIC  CONDUCTION  3 

perature,  would  reach  zero  at  —  273°C.,  as  illustrated  by  curves 
I  on  Fig.  1.     Thus,  the  resistance  may  be  expressed  by 

r  =  rQT  (1) 

where  T  is  the  absolute  temperature. 

In  alloys  of  metals  we  generally  find  a  much  lower  temperature 
coefficient,  and  find  that  the  resistance  curve  is  no  longer  a  straight 
line,  but  curved  more  or  less,  as  illustrated  by  curves  II,  Fig.  1, 
so  that  ranges  of  zero  temperature  coefficient,  as  at  A  in  curve  II, 
and  even  ranges  of  negative  temperature  coefficient,  as  at  B  in 
curve  II,  Fig.  1,  may  be  found  in  metallic  conductors  which  are 
alloys,  but  the  general  trend  is  upward.  That  is,  if  we  extend  the 
investigation  over  a  very  wide  range  of  temperature,  we  find  that 
even  in  those  alloys  which  have  a  negative  temperature  coefficient 
for  a  limited  temperature  range,  the  average  temperature  co- 
efficient is  positive  for  a  very  wide  range  of  temperature — that  is, 
the  resistance  is  higher  at  very  high  and  lower  at  very  low  tem- 
perature, and  the  zero  or  negative  coefficient  occurs  at  a  local 
flexure  in  the  resistance  curve. 

3.  The  metallic  conductors  are  the  most  important  ones  in 
industrial  electrical  engineering,  so  much  so,  that  when  speak- 
ing of  a  "conductor,"  practically  always  a  metallic  conductor  is 
understood.  The  foremost  reason  is,  that  the  resistivity  or 
specific  resistance  of  all  other  classes  of  conductors  is  so  very 
much  higher  than  that  of  metallic  conductors  that  for  directing 
the  flow  of  current  only  metallic  conductors  can  usually  come 
into  consideration. 

As,  even  with  pure  metals,  the  change  of  resistance  of  metallic 
conductors  with  change  of  temperature  is  small — about  J^  per 
cent,  per  degree  centigrade — and  the  temperature  of  most  ap- 
paratus during  their  use  does  not  vary  over  a  wide  range  of  tem- 
perature, the  resistance  of  metallic  conductors,  r,  is  usually 
assumed  as  constant,  and  the  value  corresponding  to  the  operat- 
ing temperature  chosen.  However,  for  measuring  temperature 
rise  of  electric  currents,  the  increase  of  the  conductor  resistance 
is  frequently  employed. 

Where  the  temperature  range  is  very  large,  as  between  room 
temperature  and  operating  temperature  of  the  incandescent  lamp 
filament,  the  change  of  resistance  is  very  considerable;  the  resist- 
ance of  the  tungsten  filament  at  its  operating  temperature  is  about 


4  ELECTRIC  CIRCUITS 

nine  times  its  cold  resistance  in  the  vacuum  lamp,  twelve  times  in 
the  gas-filled  lamp. 

Thus  the  metallic  conductors  are  the  most  important.  They 
require  little  discussion,  due  to  their  constancy  and  absence  of 
secondary  energy  transformation. 

Iron  makes  an  exception  among  the  pure  metals,  in  that  it  has 
an  abnormally  high  temperature  coefficient,  about  30  per  cent, 
higher  than  other  pure  metals,  and  at  red  heat,  when  approaching 
the  temperature  where  the  iron  ceases  to  be  magnetizable,  the 
temperature  coefficient  becomes  still  higher,  until  the  temperature 
is  reached  where  the  iron  ceases  to  be  magnetic.  At  this  point 
its  temperature  coefficient  becomes  that  of  other  pure  metals. 
Iron  wire — usually  mounted  in  hydrogen  to  keep  it  from  oxidizing 
— thus  finds  a  use  as  series  resistance  for  current  limitation  in 
vacuum  arc  circuits,  etc. 

Electrolytic  Conductors 

4.  The  conductors  of  the  second  class  are  the  electrolytic 
conductors.  Their  characteristic  is  that  the  conduction  is  ac- 
companied by  chemical  action.  The  specific  resistance  of  elec- 
trolytic conductors  in  general  is  about  a  million  times  higher  than 
that  of  the  metallic  conductors.  They  are  either  fused  compounds, 
or  solutions  of  compounds  in  solvents,  ranging  in  resistivity  from 
1.3  ohm-cm.,  in  30  per  cent,  nitric  acid,  and  still  lower  in  fused 
salts,  to  about  10,000  ohm-cm,  in  pure  river  water,  and  from  there 
up  to  infinity  (distilled  water,  alcohol,  oils,  etc.).  They  are  all 
liquids,  and  when  frozen  become  insulators. 

Characteristic  of  the  electrolytic  conductors  is  the  negative  tem- 
perature coefficient  of  resistance;  the  resistance  decreases  with  in- 
creasing temperature — not  in  a  straight,  but  in  a  curved  line,  as 
illustrated  by  curves  III  in  Fig.  1. 

When  dealing  with,  electrical  resistances,  in  many  cases  it  is 
more  convenient  and  gives  a  better  insight  into  the  character  of 
the  conductor,  by  not  considering  the  resistance  as  a  function  of 
the  temperature,  but  the  voltage  consumed  by  the  conductor  as  a 
function  of  the  current  under  stationary  condition.  In  this  case, 
with  increasing  current,  and  so  increasing  power  consumption, 
the  temperature  also  rises,  and  the  curve  of  voltage  for  increasing 
current  so  illustrates  the  electrical  effect  of  increasing  tempera- 
ture. The  advantage  of  this  method  is  that  in  many  cases  we  get 


ELECTRIC  CONDUCTION  5 

a  better  view  of  the  action  of  the  conductor  in  an  electric  circuit 
by  eliminating  the  temperature,  and  relating  only  electrical  quan- 
tities with  each  other.  Such  volt-ampere  characteristics  of  elec- 
tric conductors  can  easily  and  very  accurately  be  determined, 
and,  if  desired,  by  the  radiation  law  approximate  values  of  the 
temperature  be  derived,  and  therefrom  the  temperature-resist- 
ance curve  calculated,  while  a  direct  measurement  of  the  resist- 


VOLT-AMPERE  CHARACTERISTIC 

I        PURE  METALS 
II       ALLOYS 
HI      ELECTROLYTES 

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FIG.  2. 

ance  over  a  very  wide  range  of  temperature  is  extremely  difficult, 
and  often  no  more  accurate. 

In  Fig.  2,  therefore,  are  shown  such  volt-ampere  characteristics 
of  conductors.  The  dotted  straight  line  is  the  curve  of  absolutely 
constant  resistance,  which  does  not  exist.  Curves  I  and  II  show 
characteristics  of  metallic  conductors,  curve  III  of  electrolytic 
conductors.  As  seen,  for  higher  currents  I  and  II  rise  faster,  and 
III  slower  than  for  low  currents. 


6  ELECTRIC  CIRCUITS 

It  must  be  realized,  however,  that  the  volt-ampere  character- 
istic depends  not  only  on  the  material  of  the  conductor,  as  the 
temperature-resistivity  curve,  but  also  on  the  size  and  shape  of 
the  conductor,  and  its  surroundings.  For  a  long  and  thin  con- 
ductor in  horizontal  position  in  air,  it  would  be  materially  differ- 
ent numerically  from  that  of  a  short  and  thick  conductor  in  dif- 
ferent position  at  different  surrounding  temperature.  However, 
qualitatively  it  would  have  the  same  characteristics,  the  same 
characteristic  deviation  from  straight  line,  etc.,  merely  shifted  in 
their  numerical  values.  Thus  it  characterizes  the  general  nature 
of  the  conductor,  but  where  comparisons  between  different  con- 
ductor materials  are  required,  either  they  have  to  be  used  in  the 
same  shape  and  position,  when  determining  their  volt-ampere 
characteristics,  or  the  volt-ampere  characteristics  have  to  be  re- 
duced to  the  resistivity-temperature  characteristics.  The  volt- 
ampere  characteristics  become  of  special  importance  with  those 
conductors,  to  which  the  term  resistivity  is  not  physically  appli- 
cable, and  therefore  the  "effective  resistivity"  is  of  little  meaning, 
as  in  gas  and  vapor  conduction  (arcs,  etc.). 

5.  The  electrolytic  conductor  is  characterized  by  chemical 
action  accompanying  the  conduction.  This  chemical  action 
follows  Faraday's  law: 

The  amount  of  chemical  action  is  proportional  to  the  current  and 
to  the  chemical  equivalent  of  the  reaction. 

The  product  of  the  reaction  appears  at  the  terminals  or  "elec- 
trodes," between  the  electrolytic  conductor  or  "electrolyte," 
and  the  metallic  conductors.  Approximately,  0.01  mg.  of  hydro- 
gen are  produced  per  coulomb  or  ampere-second.  From  this 
electrochemical  equivalent  of  hydrogen,  all  other  chemical  reac- 
tions can  easily  be  calculated  from  atomic  weight  and  valency. 
For  instance,  copper,  with  atomic  weight  63  and  valency  2,  has 
the  equivalent  63/2  =  31.5  and  copper  therefore  is  deposited  at 
the  negative  terminal  or  "cathode,"  or  dissolved  at  the  positive 
terminal  or  "anode,"  at  the  rate  of  0.315  mg.  per  ampere-second; 
aluminum,  atomic  weight  28  and  valency  3,  at  the  rate  of  0.093 
mg.  per  ampere-second,  etc. 

The  chemical  reaction  at  the  electrodes  represents  an  energy 
transformation  between  electrical  and  chemical  energy,  and  as 
the  rate  of  electrical  energy  supply  is  given  by  current  times  vol- 
tage, it  follows  that  a  voltage  drop  or  potential  difference  occurs 
at  the  electrodes  in  the  electrolyte.  This  is  in  opposition  to  the 


ELECTRIC  CONDUCTION  7 

current,  or  a  counter  e.m.f.,  the  "counter  e.m.f.  of  electrochem- 
ical polarization,"  and  thus  consumes  energy,  if  the  chemical 
reaction  requires  energy — as  the  deposition  of  copper  from  a  solu- 
tion of  a  copper  salt.  It  is  in  the  same  direction  as  the  current, 
thus  producing  electric  energy,  if  the  chemical  reaction  produces 
energy,  as  the  dissolution  of  copper  from  the  anode. 

As  the  chemical  reaction,  and  therefore  the  energy  required  for 
it,  is  proportional  to  the  current,  the  potential  drop  at  the  elec- 
trodes is  independent  of  the  current  density,  or  constant  for  the 
same  chemical  reaction  and  temperature,  except  in  so  far  as  sec- 
ondary reactions  interfere.  It  can  be  calculated  from  the  chem- 
ical energy  of  the  reaction,  and  the  amount  of  chemical  reaction 
as  given  by  Faraday's  law.  For  instance:  1  amp.-sec.  deposits 
0.315  mg.  copper.  The  voltage  drop,  e,  or  polarization  voltage, 
thus  must  be  such  that  e  volts  times  1  amp.-sec.,  or  e  watt-sec,  or 
joules,  equals  the  chemical  reaction  energy  of  0.315  mg.  copper  in 
combining  to  the  compound  from  which  it  is  deposited  in  the 
electrolyte. 

If  the  two  electrodes  are  the  same  and  in  the  same  electrolyte 
at  the  same  temperature,  and  no  secondary  reaction  occurs,  the 
reactions  are  the  same  but  in  opposite  direction  at  the  two  elec- 
trodes, as  deposition  of  copper  from  a  copper  sulphate  solution 
at  the  cathode,  solution  of  copper  at  the  anode.  In  this  case,  the 
two  potential  differences  are  equal  and  opposite,  their  resultant 
thus  zero,  and  it  is  said  that  "no  polarization  occurs. " 

If  the  two  reactions  at  the  anode  and  cathode  are  different,  as 
the  dissolution  of  zinc  at  the  anode,  the  deposition  of  copper  at 
the  cathode,  or  the  production  of  oxygen  at  the  (carbon)  anode, 
and  the  deposition  of  zinc  at  the  cathode,  then  the  two  potential 
differences  are  unequal  and  a  resultant  remains.  This  may  be 
in  the  same  direction  as  the  current,  producing  electric  energy,  or 
in  the  opposite  direction,  consuming  electric  energy.  In  the  first 
case,  copper  deposition  and  zinc  dissolution,  the  chemical  energy 
set  free  by  the  dissolution  of  the  zinc  and  the  voltage  produced  by 
it,  is  greater  than  the  chemical  energy  consumed  in  the  deposition 
of  the  copper,  and  the  voltage  consumed  by  it,  and  the  resultant 
of  the  two  potential  differences  at  the  electrodes  thus  is  in  the 
same  direction  as  the  current,  hence  may  produce  this  current. 
Such  a  device,  then,  transforms  chemical  energy  into  electrical 
energy,  and  is  called  a  primary  cell  and  a  number  of  them,  a 
battery.  In  the  second  case,  zinc  deposition  and  oxygen  produc- 


8  ELECTRIC  CIRCUITS 

tion  at  the  anode,  the  resultant  of  the  two  potential  differences  at 
the  electrodes  is  in  opposition  to  the  current;  that  is,  the  device 
consumes  electric  energy  and  converts  it  into  chemical  energy,  as 
electrolytic  cell.  " 

Both  arrangements  are  extensively  used:  the  battery  for  pro- 
ducing electric  power,  especially  in  small  amounts,  as  for  hand 
lamps,  the  operation  of  house  bells,  etc.  The  electrolytic  cell  is 
used  extensively  in  the  industries  for  the  production  of  metals 
as  aluminum,  magnesium,  calcium,  etc.,  for  refining  of  metals  as 
copper,  etc.,  and  constitutes  one  of  the  most  important  industrial 
applications  of  electric  power. 

A  device  which  can  efficiently  be  used,  alternately  as  battery 
and  as  electrolytic  cell,  is  the  secondary  cell  or  storage  battery. 
Thus  in  the  lead  storage  battery,  when  discharging,  the  chemical 
reaction  at  the  anode  is  conversion  of  lead  peroxide  into  lead  oxide, 
at  the  cathode  the  conversion  of  lead  into  lead  oxide;  in  charging, 
the  reverse  reaction  occurs. 

6.  Specifically,  as  "polarization  cell"  is  understood  a  combina- 
tion of  electrolytic  conductor  with  two  electrodes,  of  such  char- 
acter that  no  permanent  change  occurs  during  the  passage  of  the 
current.  Such,  for  instance,  consists  of  two  platinum  electrodes 
in  diluted  sulphuric  acid.  During  the  passage  of  the  current, 
hydrogen  is  given  off  at  the  cathode  and  oxygen  at  the  anode,  but 
terminals  and  electrolyte  remain  the  same  (assuming  that  the 
small  amount  of  dissociated  water  is  replaced)  . 

In  such  a  polarization  cell,  if  e0  =  counter  e.m.f  .  of  polarization 
(corresponding  to  the  chemical  energy  of  dissociation  of  water, 
and  approximately  1.6  volts)  at  constant  temperature  and  thus 
constant  resistance  of  the  electrolyte,  the  current,  i,  is  proportional 
to  the  voltage,  e,  minus  the  counter  e.m.f.  of  polarization,  eQ: 

i  =  e-~>  (2) 


In  such  a  case  the  curve  III  of  Fig.  2  would  with  decreasing 
current  not  go  down  to  zero  volts,  but  would  reach  zero  amperes 
at  a  voltage  e  =  e0,  and  its  lower  part  would  have  the  shape  as 
shown  in  Fig.  3.  That  is,  the  current  begins  at  voltage,  e0,  and 
below  this  voltage,  only  a  very  small  "  diffusion"  current  flows. 

When  dealing  with  electrolytic  conductors,  as  when  measuring 
their  resistance,  the  counter  e.m.f.  of  polarization  thus  must  be 
considered,  and  with  impressed  voltages  less  than  the  polarization 


ELECTRIC  CONDUCTION 


9 


voltage,  no  permanent  current  flows  through  the  electrolyte,  or 
rather  only  a  very  small  " leakage"  current  or  " diffusion''  cur- 
rent, as  shown  in  Fig.  3.  When  closing  the  circuit,  however,  a 
transient  current  flows.  At  the  moment  of  circuit  closing,  no 
counter  e.m.f.  exists,  and  current  flows  under  the  full  impressed 
voltage.  This  current,  however,  electrolytically  produces  a  hy- 
drogen and  an  oxygen  film  at  the  electrodes,  and  with  their  grad- 
ual formation,  the  counter  e.m.f.  of  polarization  increases  and  de- 
creases the  current,  until  it  finally  stops  it.  The  duration  of  this 
transient  depends  on  the  resistance  of  the  electrolyte  and  on  the 
surface  of  the  electrodes,  but  usually  is  fairly  short. 

7.  This  transient  becomes  a  permanent  with  alternating  im- 
pressed voltage.     Thus,  when  an  alternating  voltage,  of  a  maxi- 


FIG.  3. 

mum  value  lower  than  the  polarization  voltage,  is  impressed 
upon  an  electrolytic  cell,  an  alternating  current  flows  through  the 
cell,  which  produces  the  hydrogen  and  oxygen  films  which  hold 
back  the  current  flow  by  their  counter  e.m.f.  The  current  thus 
flows  ahead  of  the  voltage  or  counter  e.m.f.  which  it  produces, 
as  a  leading  current,  and  the  polarization  cell  thus  acts  like  a 
condenser,  and  is  called  an  "electrolytic  condenser."  It  has  an 
enormous  electrostatic  capacity,  or  " effective  capacity,"  but  can 
stand  low  voltage  only  — 1  volt  or  less — and  therefore  is  of 
limited  industrial  value.  As  chemical  action  requires  appreciable 
time,  such  electrolytic  condensers  show  at  commercial  frequencies 
high  losses  of  power  by  what  may  be  called  "  chemical  hysteresis," 
and  therefore  low  efficiences,  but  they  are  alleged  to  become 
efficient  at  very  low  frequencies.  For  this  reason,  they  have 


10  ELECTRIC  CIRCUITS 

been  proposed  in  the  secondaries  of  induction  motors,  for  power- 
factor  compensation.  Iron  plates  in  alkaline  solution,  as  sodium 
carbonate,  are  often  considered  for  this  purpose. 

NOTE. — The  aluminum  cell,  consisting  of  two  aluminum  plates 
with  an  electrolyte  which  does  not  attack  aluminum,  often  is 
called  an  electrolytic  condenser,  as  its  current  is  leading;  that  is, 
it  acts  as  capacity.  It  is,  however,  not  an  electrolytic  condenser, 
and  the  counter  e.m.f.,  which  gives  the  capacity  effect,  is  not 
electrolytic  polarization.  The  aluminum  cell  is  a  true  electro- 
static condenser,  in  which  the  film  of  alumina,  formed  on  the 
positive  aluminum  plates,  is  the  dielectric.  Its  characteristic  is, 
that  the  condenser  is  self-healing;  that  is,  a  puncture  of  the  alum- 
ina film  causes  a  current  to  flow,  which  electrolytically  produces 
alumina  at  the  puncture  hole,  and  so  closes  it.  The  capacity  is 
very  high,  due  to  the  great  thinness  of  the  film,  but  the  energy 
losses  are  considerable,  due  to  the  continual  puncture  and  repair 
of  the  dielectric  film. 

Pyroelectric  Conductors 

8.  A  third  class  of  conductors  are  the  pyroeledric  conductors  or 
pyroelectrolytes.  In  some  features  they  are  intermediate  between 
the  metallic  conductors  and  the  electrolytes,  but  in  their  essen- 
tial characteristics  they  are  outside  of  the  range  of  either.  The 
metallic  conductors  as  well  as  the  electrolytic  conductors  give  a 
volt-ampere  characteristic  in  which,  with  increase  of  current,  the 
voltage  rises,  faster  than  the  current  in  the  metallic  conductors, 
due  to  their  positive  temperature  coefficient,  slower  than  the 
current  in  the  electrolytes,  due  to  their  negative  temperature 
coefficient. 

The  characteristic  of  the  pyroelectric  conductors,  however, 
is  such  a  very  high  negative  temperature  coefficient  of  resistance, 
that  is,  such  rapid  decrease  of  resistance  with  increase  of  tempera- 
ture, that  over  a  wide  range  of  current  the  voltage  decreases  with 
increase  of  current.  Their  volt-ampere  characteristic  thus  has  a 
shape  as  shown  diagrammatically  in  Fig.  4 — though  not  all  such 
conductors  may  show  the  complete  curve,  or  parts  of  the  curve 
may  be  physically  unattainable:  for  small  currents,  range  (1), 
the  voltage  increases  approximately  proportional  to  the  current, 
and  sometimes  slightly  faster,  showing  the  positive  temperature 
coefficient  of  metallic  conduction.  At  a  the  temperature  coeffi- 


ELECTRIC  CONDUCTION 


11 


cient  changes  from  positive  to  negative,  and  the  voltage  begins 
to  increase  slower  than  the  current,  similar  as  in  electrolytes, 
range  (2) .  The  negative  temperature  coefficient  rapidly  increases, 
and  the  voltage  rise  become  slower,  until  at  point  b  the  negative 
temperature  coefficient  has  become  so  large,  that  the  voltage  be- 
gins to  decrease  again  with  increasing  current,  range  (3).  The 
maximum  voltage  point  b  thus  divides  the  range  of  rising  charac- 
teristic (1)  and  (2),  from  that  of  decreasing  characteristic,  (3). 
The  negative  temperature  coefficient  reaches  a  maximum  and  then 
decreases  again,  until  at  point  c  the  negative  temperature  coeffi- 
cient has  fallen  so  that  beyond  this  minimum  voltage  point  c 
the  voltage  again  increases  with  increasing  current,  range  (4), 


FIG.  4. 

though  the  temperature  coefficient  remains  negative,  like  in 
electrolytic  conductors. 

In  range  (1)  the  conduction  is  purely  metallic,  in  range  (4) 
becomes  purely  electrolytic,  and  is  usually  accompanied  by 
chemical  action. 

Range  (1)  and  point  a  often  are  absent  and  the  conduction 
begins  already  with  a  slight  negative  temperature  coefficient. 

The  complete  curve,  Fig.  4,  can  be  observed  only  in  few  sub- 
stances, such  as  magnetite.  Minimum  voltage  point  c  and  range 
(4)  often  is  unattainable  by  the  conductor  material  melting  or 
being  otherwise  destroyed  by  heat  before  it  is  reached.  Such, 
for  instance,  is  the  case  with  cast  silicon.  The  maximum 
voltage  point  b  often  is  unattainable,  and  the  passage  from  range 
(2)  to  range  (3)  by  increasing  the  current  therefore  not  feasible, 


12 


ELECTRIC  CIRCUITS 


because  the  maximum  voltage  point  b  is  so  high,  that  disruptive 
discharge  occurs  before  it  is  reached.  Such  for  instance  is  the 
case  in  glass,  the  Nernst  lamp  conductor,  etc. 

9.  The  curve,  Fig.  3,  is  drawn  only  diagrammatically,  and  the 
lower  current  range  exaggerated,  to  show  the  characteristics. 
Usually  the  current  at  point  b  is  very  small  compared  with  that 
at  point  c;  rarely  more  than  one-hundredth  of  it,  and  the  actual 
proportions  more  nearly  represented  by  Fig.  5.  With  pyro- 
electric  conductors  of  very  high  value  of  the  voltage  6,  the  cur- 
rents in  the  range  (1)  and  (2)  may  not  exceed  one-millionth  of 
that  at  (3).  Therefore,  such  volt-ampere  characteristics  are 


e 

26 

X-N, 

24 

\ 

\ 

„ 

^ 

•^ 

•  2sf 

?0 

\ 

^-rf^- 

—--- 

-- 

*****" 

18 

"V>s 

14 

1? 

10 

t 

i 

!      • 

( 

{ 

J      1 

0     1 

2     1 

4     1 

6     1 

8    2 

0    2 

2    2 

4    2 

5    2 

8     3 

0    3 

2    3 

4     3 

6    3 

8    4 

)     4 

2 

FIG.  5. 

often  plotted  with  \/i  as  abscissae,  to  show  the  ranges  in  better 
proportions. 

Pyroelectric  conductors  are  metallic  silicon,  boron,  some 
forms  of  carbon  as  anthracite,  many  metallic  oxides,  especially 
those  of  the  formula  M^2)  M^  O4,  where  M(2)  is  a  bivalent, 
M(a)  a  trivalent  metal  (magnetite,  chromite),  metallic  sulphides, 
silicates  such  as  glass,  many  salts,  etc. 

Intimate  mixtures  of  conductors,  as  graphite,  coke,  powdered 
metal,  with  non-conductors  as  clay,  carborundum,  cement,  also 
have  pyroelectric  conduction.  Such  are  used,  for  instance,  as 
'  'resistance  rods"  in  lightning  arresters,  in  some  rheostats,  as 


ELECTRIC  CONDUCTION 


13 


cement  resistances  for  high-frequency  power  dissipation  in  re- 
actances, etc.  Many,  if  not  all  so-called  "insulators"  probably 
are  in  reality  pyroelectric  conductors,  in  which  the  maximum 
voltage  point  b  is  so  high,  that  the  range  (3)  of  decreasing  charac- 
teristic can  be  reached  only  by  the  application  of  external  heat, 
as  in  the  Nernst  lamp  conductor,  or  can  not  be  reached  at  all, 
because  chemical  dissociation  begins  below  its  temperature,  as 
in  organic  insulators. 

Fig.  6  shows  the  volt-ampere  characteristics  of  two  rods  of 
cast  silicon,  10  in.  long  and  0.22  in.  in  diameter,  with  \A  as  ab- 


VOLT-AMPERE  CHARACTERISTIC 
OF  CAST  SILICON 


FIG.  6. 

scissse  and  Fig.  7  their  approximate  temperature-resistance 
characteristics.  The  curve  II  of  Fig.  7  is  replotted  in  Fig.  8, 
with  log  r  as  ordinates.  Where  the  resistivity  varies  over  a  very 
wide  range,  it  often  is  preferable  to  plot  the  logarithm  of  the 
resistivity.  It  is  interesting  to  note  that  the  range  (3)  of  curve 
II,  between  700°  and  1400°,  is  within  the  errors  of  observation 
represented  by  the  expression 


r  =  O.QIE 


9080 
T~ 


where  T  is  the  absolute  temperature  (— 273°C.  as  zero  point). 
The  difference  between  the  two  silicon  rods  is,  that  the  one  con- 


14 


ELECTRIC  CIRCUITS. 


tains  1.4  per  cent.,  the  other  only  0.1  per  cent,  carbon;  besides 
this,  the  impurities  are  less  than  1  per  cent. 

As  seen,  in  these  silicon  rods  the  r^nge  (4)  is  not  yet  reached  at 
the  melting  point. 

Fig.  9  shows  the  volt-ampere  characteristic,  with  \/f  as  abscis- 
sae, and  Fig.  10  the  approximate  resistance  temperature  char- 


acteristic derived  therefrom,  with  log  r  as  ordinates,  of  a  magnetic 
rod  6  in.  long  and  %  in.  in  diameter,  consisting  of  90  per  cent, 
magnetite  (Fe3O4),  9  per  cent,  chromite  (FeCr2O4)  and  1  per  cent, 
sodium  silicate,  sintered  together. 

10.  As  result  of  these  volt-ampere  characteristics,  Figs.  4  to 
10,  pyroelectric  conductors  as  structural  elements  of  an  electric 
circuit  show  some  very  interesting  effects,  which  may  be  illus- 


ELECTRIC  CONDUCTION 


15 


t rated  on  the  magnetite  rod,  Fig.  9.  The  maximum  terminal  vol- 
tage, which  can  exist  across  this  rod  in  stationary  conditions,  is 
25  volts  at  1  amp.  With  increasing  terminal  voltage,  the  current 
thus  gradually  increases,  until  25  volts  is  reached,  and  then  with- 
out further  increase  of  the  impressed  voltage  the  current  rapidly 
rises  to  short-circuit  values.  Thus,  such  resistances  can  be  used 
as  excess-voltage  cutout,  or,  when  connected  between  circuit  and 
ground,  as  excess-voltage  grounding  device:  below  24  volts,  it 


\ 

RESISTIVITY    TEMPERATURE 
CHARACTERISTIC  OF 
CAST    SILICON    ROD 
25  CM.  LENGTH  0.56  CM.  DIAMETER 
DOTTED  CURVE    r=0.0l£  *¥* 

LOG 
2.8 

.T. 

•          — 

2.6 



—  —  . 

--—  ^ 

\ 

2.4 

X 

\ 

\ 

2.2 

\ 

^ 

?0 

S 

18 

k 

1,6 

\ 

V 

1,4 

\ 

1.3 

\ 

LO 

s 

\ 

0.8 

\ 

\ 

O.fi 

\ 

V 

04 

X 

\ 

0,? 

D 

EGRI 

EES 

c. 

I 

X)    2( 

0    3C 

K)     4( 

)0    5( 

X)    ft 

K>    7( 

K)    800    9C 

0    10 

00  11 

00  12 

00  13 

X)  14 

00 

FIG.  8. 

bypasses  a  negligible  current  only,  but  if  the  voltage  rises  above 
25  volts,  it  short-circuits  the  voltage  and  so  stops  a  further  rise,  or 
operates  the  circuit-breaker,  etc.  As  the  decrease  of  resistance  is 
the  result  of  temperature  rise,  it  is  not  instantaneous;  thus  the  rod 
does  not  react  on  transient  voltage  rises,  but  only  on  lasting  ones. 
Within  a  considerable  voltage  range — between  16  and  25  volts 
— three  values  of  current  exist  for  the  same  terminal  voltage. 
Thus  at  20  volts  between  the  terminals  of  the  rod  in  Fig.  9,  the 
current  may  be  0.02  amp.,  or  4.1  amp.,  or  36  amp.  That  is,  in 


scries  in  a  constant-current  circuit  of  4.1  amp.  this  rod  would 
show  the  same  terminal  voltage  as  in  a  0.02-amp.  or  a  36-amp. 
constant-current  circuit,  20  volts.  On  constant-potential  supply, 
however,  only  the  range  (1)  and  (2),  and  the  range  (4)  is  stable, 
but  the  range  (3)  is  unstable,  and  hero  we  have  a  conductor,  which 
is  unstable  in  a  certain  range  of  currents,  from  point  6  at  1  amp. 
to  point  c  at  20  amp.  At  20  volts  impressed  upon  the  rod,  0.02 
amp.  may  pass  through  it,  and  the  conditions  are  stable.  That 
is,  a  tendency  to  increase  of  current  would  check  itself  by  requir- 
ing an  increase  of  voltage  beyond  that  supplied,  and  a  decrease  of 


, 

; 

I 

2 

5 

: 

•> 

•\ 

9 

c 

4 

AM 

PER 

ES- 

-+• 

X 

vo 

LFb 
20 

> 

^>  — 

-^s 

/ 

\ 

X 

tx 

1 

\ 

A 
\ 

V 

/ 

20 

-^ 

7* 

/ 

-18- 

Ifi 

*"^, 

—  — 

10 

VOLT-  AMPERE    CHARACTERISTIC 
OF 
MAGNETITE  RESISTANCE 

10 

-4- 

\/A 

rfPE 

Eli 

->- 

j 

! 

. 

( 

1 

5 

Fia.  9. 

current  would  reduce  the  voltage  consumption  below  that  em- 
ployed, and  thus  be  checked.  At  the  same  impressed  20  volts, 
36  amp.  may  pass  through  the  rod — or  1800  times  as  much  as 
before — and  the  conditions  again  are  stable.  A  current  of  4.1 
amp.  also  would  consume  a  terminal  voltage  of  20,  but  the  condi- 
tion now  is  unstable;  if  the  current  increases  ever  so  little,  by  a 
momentary  voltage  rise,  then  the  voltage  consumed  by  the  rod 
decreases,  becomes  less  than  the  terminal  voltage  of  20,  and 
the  current  thus  increases  by  the  supply  voltage  exceeding  the 
consumed  voltage.  This,  however,  still  further  decreases  the 


CONDUCTION 


17 


consumed  vollag<»  and  thereby  increases  the  current,  and  the  cur- 
rent  rapidly  rises,  until  conditions  become  stable  at  36  amp.  In- 
versely, a  niomenlary  decrease  of  the  current  below  4.1  amp.  in- 
creases UK;  voltage  required  by  UK;  rod,  and  this  higher  voltage;  not 
being  available  at  constant  supply  voltage,  the  current  decreases. 


^ 

\TUF 

3°F 

RLSIiiTIYITY  ThMf'ERj 
CHARACTERISTIC 
MAGNETITE  ROI 

15  x  1,9  CM; 

<E 

L( 

G  r 
fl6 

\ 

5U 

\ 

/ 

«?! 

\ 

a;o 

\ 

8,8 

K« 

— 

8^4 

\ 

Aft 

\ 

H.O 

V 

V 

TR 

\ 

7,6 

\ 

T4 

\ 

7,2 

^^>N 

^^ 

^^-,   ! 

TO 

«R 

. 

| 

EGR 

EES 

c. 

1( 

10     2 

)0    3( 

)0    4 

)0    & 

)0    G 

K)    7 

X)  a 

X)    ix 

X)    10 

00  11 

00  11 

!00 

Fio.  10. 

This,  however,  still  further  increases  the  required  voltage  and 
decreases  the  current,  until  conditions  become  stable  at  0.02  amp. 
With  the  silicon  rod  II  of  Fig.  6,  on  constant-potential  supply, 
with  increasing  voltage  the  current  and  the  temperature  increases 
gr.-i dually,  until  57.5  volts  are  reached  at  about  450°C.;  then, 
without  further  voltage  increase,  current  and  temperature  rapidly 
increase  until  the  rod  melts.    Thus: 
2 


18  ELECTRIC  CIRCUITS 

Condition  of  stability  of  a  conductor  on  constant-voltage  sup- 
ply is,  that  the  volt-ampere  characteristic  is  rising,  that  is,  an  in- 
crease of  current  requires  an  increase  of  terminal  voltage. 

A  conductor  with  falling  volt-ampere  characteristic,  that  is,  a 
conductor  in  which  with  increase  of  current  the  terminal  voltage 
decreases,  is  unstable  on  constant-potential  supply. 

11.  An  important  application  of  pyroelectric  conduction  has 
been  the  glower  of  the  Nernst  lamp,  which  before  the  develop- 
ment of  the  tungsten  lamp  was  extensively  used  for  illumination. 

Pyroelectrolytes  cover  the  widest  range  of  conductivities;  the 
alloys  of  silicon  with  iron  and  other  metals  give,  depending  on 
their  composition,  resistivities  from  those  of  the  pure  metals  up  to 
the  lower  resistivities  of  electrolytes:  1  ohm  per  cm.3;  borides, 
carbides,  nitrides,  oxides,  etc.,  gave  values  from  1  ohm  per  cm.3 
or  less,  up  to  megohms  per  cm.3,  and  gradually  merge  into  the 
materials  which  usually  are  classed  as  "insulators." 

The  pyroelectric  conductors  thus  are  almost  the  only  ones 
available  in  the  resistivity  range  between  the  metals,  0.0001  ohm- 
cm,  and  the  electrolytes,  1  ohm-cm. 

Pyroelectric  conductors  are  industrially  used  to  a  considerable 
extent,  since  they  are  the  only  solid  conductors,  which  have  re- 
sistivities much  higher  than  metallic  conductors.  In  most  of  the 
industrial  uses,  however,  the  dropping  volt-ampere  characteristic 
is  not  of  advantage,  is  often  objectionable,  and  the  use  is  limited 
to  the  range  (1)  and  (2)  of  Fig.  3.  It,  therefore,  is  of  importance 
to  realize  their  pyroelectric  characteristics  and  the  effect  which 
they  have  when  overlooked  beyond  the  maximum  voltage  point. 
Thus  so-called  "graphite  resistances"  or  "carborundum  resist- 
ances/ '  used  in  series  to  lightning  arresters  to  limit  the  discharge, 
when  exposed  to  a  continual  discharge  for  a  sufficient  time  to 
reach  high  temperature,  may  practically  short-circuit  and  there- 
by fail  to  limit  the  current. 

12.  From  the  dropping  volt-ampere  characteristic  in  some 
pyroelectric  conductors,  especially  those  of  high  resistance,  of 
very  high  negative  temperature  coefficient  and  of  considerable 
cross-section,  results  the  tendency  to  unequal  current  distribution 
and  the  formation  of  a  "luminous  streak,"  at  a  sudden  applica- 
tion of  high  voltage.     Thus,  if  the  current  passing  through  a 
graphite-clay  rod  of  a  few  hundred  ohms  resistance  is  gradually 
increased,  the  temperature  rises,  the  voltage  first  increases  and 
then  decreases,  while  the  rod  passes  from  range  (2)  into  the 


ELECTRIC  CONDUCTION  19 

range  (3)  of  the  volt-ampere  characteristic,  but  the  temperature 
and  thus  the  current  density  throughout  the  section  of  the  rod  is 
fairly  uniform.  If,  however,  the  full  voltage  is  suddenly  applied, 
such  as  by  a  lightning  discharge  throwing  line  voltage  on  the 
series  resistances  of  a  lightning  arrester,  the  rod  heats  up  very 
rapidly,  too  rapidly  for  the  temperature  to  equalize  throughout  the 
rod  section,  and  a  part  of  the  section  passes  the  maximum  voltage 
point  6  of  Fig.  4  into  the  range  (3)  and  (4)  of  low  resistance,  high 
current  and  high  temperature,  while  most  of  the  section  is  still  in 
the  high-resistance  range  (2)  and  never  passes  beyond  this  range, 
as  it  is  practically  short-circuited.  Thus,  practically  all  the  cur- 
rent passes  by  an  irregular  luminous  streak  through  a  small  sec- 
tion of  the  rod,  while  most  of  the  section  is  relatively  cold  and 
practically  does  not  participate  in  the  conduction.  Gradually, 
by  heat  conduction  the  temperature  and  the  current  density  may 
become  more  uniform,  if  before  this  the  rod  has  not  been  de- 
stroyed by  temperature  stresses.  Thus,  tests  made  on  such  con- 
ductors by  gradual  application  of  voltage  give  no  information  on 
their  behavior  under  sudden  voltage  application.  The  liability  to 
the  formation  of  such  luminous  streaks  naturally  increases  with 
decreasing  heat  conductivity  of  the  material,  and  with  increasing 
resistance  and  temperature  coefficient  of  resistance,  and  with  con- 
ductors of  extremely  high  temperature  coefficient,  such  as  silicates, 
oxides  of  high  resistivity,  etc.,  it  is  practically  impossible  to  get 
current  to  flow  through  any  appreciable  conductor  section,  but 
the  conduction  is  always  streak  conduction. 

Some  pyroelectric  conductors  have  the  characteristic  that  their 
resistance  increases  permanently,  often  by  many  hundred  per 
cent,  when  the  conductor  is  for  some  time  exposed  to  high-fre- 
quency electrostatic  discharges. 

Coherer  action,  that  is,  an  abrupt  change  of  conductivity  by  an 
electrostatic  spark,  a  wireless  wave,  etc.,  also  is  exhibited  by  some 
pyroelectric  conductors. 

13.  Operation  of  pyroelectric  conductors  on  a  constant-voltage 
circuit,  and  in  the  unstable  branch  (3) ,  is  possible  by  the  insertion 
of  a  series  resistance  (or  reactance,  in  alternating-current  circuits) 
of  such  value,  that  the  resultant  volt-ampere  characteristic  is 
stable,  that  is,  rises  with  increase  of  current.  Thus,  the  con- 
ductor in  Fig.  4,  shown  as  I  in  Fig.  11,  in  series  with  the  metallic 
resistance  giving  characteristic  A ,  gives  the  resultant  characteris- 
tic II  in  Fig.  11,  which  is  stable  over  the  entire  range.  /  in  series 


20 


ELECTRIC  CIRCUITS 


with  a  smaller  resistance,  of  characteristic  B,  gives  the  resultant 
characteristic  ///.  In  this,  the  unstable  range  has  contracted  to 
from  bf  to  c'.  Further  discussion  of  the  instability  of  such  con- 
ductors, the  effect  of  resistance  in  stablizing  them,  and  the  result- 


STABILITY  CURVES 

OF 
PYRO  ELECTRIC  CONDUCTOR 


A/ 


X 


r 


ir 


26, 

2 


1.0      11      12      13      1,4      15 


FIG.  11. 


ant  " stability  curve"  are  found  in  the  chapter  on  "Instability 
of  Electric  Circuits,"  under  "Arcs  and  Similar  Conductors." 

14.  It  is  doubtful  whether  the  pyroelectric  conductors  really 
form  one  class,  or  whether,  by  the  physical  nature  of  their  conduc- 
tion, they  should  not  be  divided  into  at  least  two  classes: 

1.  True  pyroelectric  conductors,  in  which  the  very  high  nega- 
tive temperature  coefficient  is  a  characteristic  of  the  material. 


ELECTRIC  CONDUCTION 


21 


In  this  class  probably  belong  silicon  and  its  alloys,  boron,  mag- 
netite and  other  metallic  oxides,  sulphides,  carbides,  etc. 

2.  Conductors  which  are  mixtures  of  materials  of  high  conduc- 
tivity, and  of  non-conductors,  and  derive  their  resistance  from 
the  contact  resistance  between  the  conducting  particles  which 
are  separated  by  non-conductors.  As  contact  resistance  shares 
with  arc  conduction  the  dropping  volt-ampere  characteristic, 
such  mixtures  thereby  imitate  pyroelectric  conduction.  In  this 
class  probably  belong  the  graphite-clay  rods  industrially  used. 
Powders  of  metals,  graphite  and  other  good  conductors  also 
belong  in  this  class. 

The  very  great  increase  of  resistance  of  some  conductors  under 
electrostatic  discharges  probably  is  limited  to  this  class,  and  is 
the  result  of  the  high  current  density  of  the  condenser  discharge 
burning  off  the  contact  points. 

Coherer  action  probably  is  limited  also  to  those  conductors,  and 
is  the  result  of  the  minute  spark  at  the  contact  points  initiating 
conduction. 

Carbon 

15.  In  some  respects  outside  of  the  three  classes  of  conductors 
thus  far  discussed,  in  others  intermediate  between  them,  is  one  of 


VOLT-AMPERE  CHARACTERISTIC 
OF  CARBON 

/, 

/ 

// 

'/ 

/ 

/  ' 
/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

\ 

/ 

/ 

/ 

r 

/ 

/ 

(fl 

/ 

/ 

o 

> 

V 

/ 

/ 

/ 

/* 

/ 

S 

/ 

/ 

'n 

/ 

J^ 

t 

/ 

/ 

/ 

^ 

7 

^ 

~? 

A 

MPE 

RE 

5  — 

-> 

£ 

^ 

FIG.  12. 


the  industrially  most  important  conductors,  carbon.     It  exists  in  a 
large  variety  of  modifications  of  different  resistance  characteris- 


22 


ELECTRIC  CIRCUITS 


tics,  which  all  are  more  or  less  intermediate  between  three  typical 
forms : 

1.  Metallic  Carbon. — It  is  produced  from  carbon  deposited  on 
an  incandescent  filament,  from  hydrocarbon  vapors  at  a  partial 
vacuum,  by  exposure  to  the  highest  temperatures  of  the  electric 
furnace.  Physically,  it  has  metallic  characteristics:  high  elas- 


-24 


RESISTANCE-TEMPERATURE 

CHARACTERISTIC    OF    CARBON 

RESISTIVITY    IN    OHM-CENTIMETERS 


100  200  300  400  500  COO 


AP 


:PERATURE  c 


H- 


fc--7- 


1.0 


.8_ 


.4- 


900  1000 1100 1200 1300 1400  1500  1600 1700 


FIG.  13. 

ticity,  metallic  luster,  etc.,  and  electrically  it  has  a  relatively 
low  resistance  approaching  that  of  metallic  conduction,  and  a 
positive  temperature  coefficient  of  resistance,  of  about  0.1  per 
cent,  per  degree  C. — that  is,  of  the  same  magnitude  as  mercury 
or  cast  iron. 

The  coating  of  the  "Gem"  filament  incandescent  lamp  con- 
sists of  this  modification  of  carbon. 


ELECTRIC  CONDUCTION  23 

2.  Amorphous  carbon,  as  produced  by  the  carbonization  of 
cellulose.     In  its  purest  form,  as  produced  by  exposure  to  the 
highest  temperatures  of  the  electric  furnace,  it  is  characterized  by 
a  relatively  high  resistance,  and  a  negative  temperature   coeffi- 
cient of  resistance,  its  conductivity  increasing  by  about  0. 1  per 
cent,  per  degree  C. 

3.  Anthracite. — It  has  an  extremely  high  resistance,  is  prac- 
tically an  insulator,  but  has  a  very  high  negative  temperature 
coefficient  of  resistance,  and  thus  becomes  a  fairly  good  conductor 
at  high  temperature,  but  its  heat  conductivity  is  so  low,  and  the 
negative  temperature  coefficient  of  resistance  so  high,  that  the 
conduction  is  practically  always  streak  conduction,  and  at  the 
high  temperature  of  the  conducting  luminous  streak,  conversion 
to  graphite  occurs,  with  a  permanent  decrease  of  resistance. 

(1)  thus  shows  the  characteristics  of  metallic  conduction,  (2) 
those  of  electrolytic  conduction,  and  (3)  those  of  pyroelectric 
conduction. 

Fig.  12  shows  the  volt-ampere  characteristics,  and  Fig.  13  the 
resistance-temperature  characteristics  of  amorphous  carbon — 
curve  I — and  metallic  carbon —  curve  II. 

Insulators 

16.  As  a  fourth  class  of  conductors  may  be  considered  the  so- 
called  "  insulators, "  that  is,  conductors  which  have  such  a  high 
'  specific  resistance,  that  they  can  not  industrially  be  used  for  con- 
veying electric  power,  but  on  the  contrary  are  used  for  restraining 
the  flow  of  electric  power  to  the  conductor,  or  path,  by  separating 
the  conductor  from  the  surrounding  space  by  such  an  insulator. 
The  insulators  also  have  a  conductivity,  but  their  specific  resist- 
ance is  extremely  high.  For  instance,  the  specific  resistance  of 
fiber  is  about  1012,  of  mica  10 14,  of  rubber  1016  ohm-cm.,  etc. 

As,  therefore,  the  distinction  between  conductor  and  insulator 
is  only  qualitative,  depending  on  the  application,  and  more  par- 
ticularly on  the  ratio  of  voltage  to  current  given  by  the  source  of 
power,  sometimes  a  material  may  be  considered  either  as  insulator 
or  as  conductor.  Thus,  when  dealing  with  electrostatic  machines, 
which  give  high  voltages,  but  extremely  small  currents,  wood, 
paper,  etc.,  are  usually  considered  as  conductors,  while  for  the 
low-voltage  high-current  electric  lighting  circuits  they  are  insula- 
tors, and  for  the  high-power  very  high- voltage  transmission  cir- 


24  ELECTRIC  CIRCUITS 

cults  they  are  on  the  border  line,  are  poor  conductors  and  poor 
insulators. 

Insulators  usually,  if  not  always,  have  a  high  negative  tempera- 
ture coefficient  of  resistance,  and  the  resistivity  often  follows 
approximately  the  exponential  law, 

aT  (3) 


where  T  =  temperature.  That  is,  the  resistance  decreases  by  the 
same  percentage  of  its  value,  for  every  degree  C.  For  instance, 
it  decreases  to  one-tenth  for  every  25°C.  rise  of  temperature,  so 
that  at  100°C.  it  is  10,000  times  lower  than  at  0°C.  Some  tem- 
perature-resistance curves,  with  log  r  as  ordinates,  of  insulating 
materials  are  given  in  Fig.  14. 

As  the  result  of  the  high  negative  temperature  coefficient,  for  a 
sufficiently  high  temperature,  the  insulating  material,  if  not  de- 
stroyed by  the  temperature,  as  is  the  case  with  organic  materials, 
becomes  appreciably  conducting,  and  finally  becomes  a  fairly 
good  conductor,  usually  an  electrolytic  conductor. 

Thus  the  material  of  the  Nernst  lamp  (rare  oxides,  similar  to 
the  Welsbach  mantle  of  the  gas  industry),  is  a  practically  perfect 
insulator  at  ordinary  temperatures,  but  becomes  conducting  at 
high  temperature,  and  is  then  used  as  light-giving  conductor. 

Fig.  15  shows  for  a  number  of  high-resistance  insulat- 
ing materials  the  temperature-resistance  curve  at  the  range 
where  the  resistivity  becomes  comparable  with  that  of  other 
conductors. 

17.  Many  insulators,  however,  more  particularly  the  organic 
materials,  are  chemically  or  physically  changed  or  destroyed, 
before  the  temperature  of  appreciable  conduction  is  reached, 
though  even  these  show  the  high  negative  temperature  coefficient. 
With  some,  as  varnishes,  etc.,  the  conductivity  becomes  sufficient, 
at  high  temperatures,  though  still  below  carbonization  tempera- 
ture, that  under  high  electrostatic  stress,  as  in  the  insulation  of 
high-  voltage  apparatus,  appreciable  energy  is  represented  by  the 
leakage  current  through  the  insulation,  and  in  this  case  rapid 
izr  heating  and  final  destruction  of  the  material  may  result. 
That  is,  such  materials,  while  excellent  insulators  at  ordinary 
temperature,  are  unreliable  at  higher  temperature. 

It  is  quite  probable  that  there  is  no  essential  difference  between 
the  true  pyroelectric  conductors,  and  the  insulators,  but  the  latter 
are  merely  pyroelectric  conductors  in  which  the  initial  resistivity 


ELECTRIC  CONDUCTION 


25 


and  the  voltage  at  the  maximum  point  b  are  so  high,  that  the 
change  from  the  range  (2)  of  the  pyroelectrolyte,  Fig.  4,  to  the 
range  (3)  can  not  be  produced  by  increase  of  voltage.  That  is, 
the  distinction  between  pyroelectric  conductor  and  insulator 
would  be  the  quantitative  .one,  that  in  the  former  the  maximum 


RESISTIVITY-TEMPERATURE 

CHARACTERISTICS  OF 

INSULATORS 


FIG.  14. 

voltage  point  of  the  volt-ampere  characteristic  is  within  experi- 
mental reach,  while  with  the  latter  it  is  beyond  reach. 

Whether  this  applies  to  all  insulators,  or  whether  among  or- 
ganic compounds  as  oils,  there  are  true  insulators,  which  are  not 
pyroelectric  conductors,  is  uncertain. 


26 


ELECTRIC  CIRCUITS 


Positive  temperature  coefficient  of  resistivity  is  very  often  met 
in  insulating  materials  such  as  oils,  fibrous  materials,  etc.  In  this 
case,  however,  the  rise  of  resistance  at  increase  of  temperature 
usually  remains  permanent  after  the  temperature  is  again  lowered, 


\ 

\ 

RESISTIVITY-TEMPERATURE 
CHARACTERISTIC  OF 
HIGH    TEMPERATURE    INSULATORS 

_10_ 
9.5 

\ 

\ 

LOV 

'ER 

.IMIT 

OF 

\ 

9 

INS 

JLAT 

NUC 

ILS 

s 

V 

8.5 

\ 

8 

\ 

7.5 

\ 

\ 

7 

\ 

\ 

BO 

?ON 

NITR 

D 

6  5 

v 

\ 

\ 

\ 

g 

\\ 

\ 

V^ 

\ 

FUS 

ED   r 

/1AGN 

ESIA 

r>& 

\ 

X 

s^x^ 

^L 

«• 

5 

\ 


^  f 

ORC 

:LAU 

1 

45 

PUF 

E  Rl 

/ER 

s 

REG 

ONS1 

'RUG 

TED 

LAVA 

4 

I 

VATE 

R 

35 

3 

?,5 

| 

1  5 

1 

5 

1C 

0     2( 

X)     3( 

K)     4( 

10     & 

K)    6( 

K)     7( 

)0     8t 

K)     & 

K)    1C 

00  '( 

FIG.  15. 


and  the  apparent  positive  temperature  coefficient  was  due  to  the 
expulsion  of  moisture  absorbed  by  the  material.  With  insulators 
of  very  high  resistivity,  extremely  small  traces  of  moisture  may 
decrease  the  resistivity  many  thousandfold,  and  the  conductivity 
of  insulating  materials  very  often  is  almost  entirely  moisture  con- 


ELECTRIC  CONDUCTION  27 

duction,  that  is,  not  due  to  the  material  proper,  but  due  to  the 
moisture  absorbed  by  it.  In  such  a  case,  prolonged  drying  may 
increase  the  resistivity  enormously,  and  when  dry,  the  material 
then  shows  the  negative  temperature  coefficient  of  resistance, 
incident  to  pyroelectric  conduction. 


CHAPTER  II 

ELECTRIC  CONDUCTION.     GAS  AND  VAPOR 
CONDUCTORS 

Gas,  Vapor  and  Vacuum  Conduction 

18.  As  further,  and  last  class  may  be  considered  vapor,  gas 
and  vacuum  conduction.  Typical  of  this  is,  that  the  volt-ampere 
characteristic  is  dropping,  that  is,  the  voltage  decreases  with  in- 
crease of  current,  and  that  luminescence  accompanies  the  con- 
duction, that  is,  conversion  of  electric  energy  into  light. 

Thus,  gas  and  vapor  conductors  are  unstable  on  constant- 
potential  supply,  but  stable  on  constant  current.  On  constant 
potential  they  require  a  series  resistance  or  reactance,  to  produce 
stability. 

Such  conduction  may  be  divided  into  three  distinct  types: 
spark  conduction,  arc  conduction,  and  true  electronic  conduction. 

In  spark  conduction,  the  gas  or  vapor  which  fills  the  space  be- 
tween the  electrodes  is  the  conductor.  The  light  given  by  the 
gaseous  conductor  thus  shows  the  spectrum  of  the  gas  or  vapor 
which  fills  the  space,  but  the  material  of  the  electrodes  is  imma- 
terial, that  is,  affects  neither  the  light  nor  the  electric  behavior  of 
the  gaseous  conductor,  except  indirectly,  in  so  far  as  the  section 
of  the  conductor  at  the  terminals  depends  upon  the  terminal  sur- 
face. 

In  arc  conduction,  the  conductor  is  a  vapor  stream  issuing  from 
the  negative  terminal  or  cathode,  and  moving  toward  the  anode 
at  high  velocity.  The  light  of  the  arc  thus  shows  the  spectrum 
of  the  negative  terminal  material,  but  not  that  of  the  gas  in  the 
surrounding  space,  nor  that  of  the  positive  terminal,  except  indi- 
rectly, by  heat  luminescence  of  material  entering  the  arc  con- 
ductor from  the  anode  or  from  surrounding  space. 

In  true  electronic  conduction,  electrons  existing  in  the  space, 
or  produced  at  the  terminals  (hot  cathode),  are  the  conductors. 
Such  conduction  thus  exists  also  in  a  perfect  vacuum,  and  may  be 
accompanied  by  practically  no  luminescence. 

28 


ELECTRIC  CONDUCTION  29 

Disruptive  Conduction 

19.  Spark  conduction  at  atmospheric  pressure  is  the  disruptive 
spark,  streamers,  and  corona.  In  a  partial  vacuum,  it  is  the 
Geissler  discharge  or  glow  discharge.  Spark  conduction  is  dis- 
continuous, that  is,  up  to  a  certain  voltage,  the  "disruptive 
voltage,"  no  conduction  exists,  except  perhaps  the  extremely 
small  true  electronic  conduction.  At  this  voltage  conduction 
begins  and  continues  as  long  as  the  voltage  persists,  or,  if  the 
source  of  power  is  capable  of  maintaining  considerable  current, 
the  spark  conduction  changes  to  arc  conduction,  by  the  heat  de- 
veloped at  the  negative  terminal  supplying  the  conducting  arc 
vapor  stream.  The  current  usually  is  small  and  the  voltage 
high.  Especially  at  atmospheric  pressure,  the  drop  of  the  volt- 
ampere  characteristic  is  extremely  steep,  so  that  it  is  practically 
impossible  to  secure  stability  by  series  resistance,  but  the  con- 
duction changes  to  arc  conduction,  if  sufficient  current  is  avail- 
able, as  from  power  generators,  or  the  conduction  ceases  by  the 
voltage  drop  of  the  supply  source,  and  then  starts  again  by  the 
recovery  of  voltage,  as  with  an  electrostatic  machine.  Thus 
spark  conduction  also  is  called  disruptive  conduction  and  discon- 
tinuous conduction. 

Apparently  continuous — though  still  intermittent — spark  con- 
duction is  produced  at  atmospheric  pressure  by  capacity  in  series 
to  the  gaseous  conductor,  on  an  alternating-voltage  supply,  as 
corona,  and  as  Geissler  tube  conduction  at  a  partial  vacuum,  by 
an  alternating-supply  voltage  with  considerable  reactance  or 
resistance  in  series,  or  from  a  direct-current  source  of  very  high 
voltage  and  very  limited  current,  as  an  electrostatic  machine. 

In  the  Geissler  tube  or  vacuum  tube,  on  alternating-voltage 
supply,  the  effective  voltage  consumed  by  the  tube,  at  constant 
temperature  and  constant  gas  pressure,  is  approximately  con- 
stant and  independent  of  the  effective  current,  that  is,  the  volt- 
ampere  characteristic  a  straight  horizontal  line.  The  Geissler 
tube  thus  requires  constant  current  or  a  steadying  resistance  or 
reactance  for  its  operation.  The  voltage  consumed  by  the  Geiss- 
ler tube  consists  of  a  potential  drop  at  the  terminals,  the  "termi- 
nal drop, "  and  a  voltage  consumed  in  the  luminous  stream,  the 
"stream  voltage."  Both  greatly  depend  on  the  gas  pressure, 
and  vary,  with  changing  gas  pressure,  in  opposite  directions :  the 
terminal  drop  decreases  and  the  stream  voltage  increases  with 
increasing  gas  pressure,  and  the  total  voltage  consumed  by  the 


30 


ELECTRIC  CIRCUITS 


tube  thus  gives  a  minimum  at  some  definite  gas  pressure.     This 
pressure  of  minimum  voltage  depends  on  the  length  of  the  tube, 


FIG.  16. 


.01 

0 

1 

L 

0 

5000 

mnr 

HG 

'RES! 

URE, 

p 

y 

1 

\ 

/ 

/ 

4000 

N 

rOTAl 
^^,, 

VOL 

TAGE 

^ 

y 

/ 

3500 

\ 

0 

.1  AN 

3  0.0! 

AMP 

#/ 

/ 

CO 

-3000 
2500 

\ 

A 

7 

0 

> 

\ 

A 

^ 

2000 

>< 

fa. 

ME 

RCUR 

y  VA 

'OR 

1500- 

/ 

•^0 

1  AM 

>.x 

^ 

*OF> 

1000 

**•  — 

"••              i    i 

Kflfl 

8.0 

LOG 
9 

P 

o 

o 

o 

FIG.  17. 


and  the  longer  the  tube,  the  lower  is  the  gas  pressure  which  gives 
minimum  total  voltage. 


ELECTRIC  CONDUCTION  31 

Fig.  16  shows  the  voltage-pressure  characteristic,  at  constant 
current  of  0.1  amp.  and  0.05  amp.,  of  a  Geissler  tube  of  1.3  cm. 
internal  diameter  and  200  cm.  length,  using  air  as  conductor,  and 
Fig.  17  the  characteristic  of  the  same  tube  with  mercury  vapor  as 
conductor.  Figs.  16  and  17  also  show  the  two  component  voltages, 
the  terminal  drop  and  the  stream  voltage,  separately.  As  ab- 
scissae are  used  the  log  of  the  gas  pressure,  in  millimeter  mercury 
column.  As  seen,  the  terminal  drop  decreases  with  increasing 
gas  pressure,  and  becomes  negligible  compared  with  the  stream 
voltage,  at  atmospheric  pressure. 

The  voltage  gradient,  per  centimeter  length  of  stream,  varies 
from  5  to  20  volts,  at  gas  or  vapor  pressure  from  0.06  to  0.9 
mm.  At  atmospheric  pressure  (760  mm.)  the  disruptive  voltage 
gradient,  which  produces  corona,  is  21,000  volts  effective  per 
centimeter.  The  specific  resistance  of  the  luminous  stream  is  from 
65  to  500  ohms  per  cm.3  in  the  Geissler  tube  conduction  of  Figs. 
16  and  17 — though  this  term  has  little  meaning  in  gas  conduction. 
The  specific  resistance  of  the  corona  in  air,  as  it  appears  on  trans- 
mission lines  at  very  high  'voltages,  is  still  very  much  higher. 

Arc  Conduction 

20.  In  the  electric  arc,  the  current  is  carried  across  the  space 
between  the  electrodes  or  arc  terminals  by  a  stream  of  electrode 
vapor,  which  issues  from  a  spot  on  the  negative  terminal,  the 
so-called  cathode  spot,  as  a  high-velocity  blast  (probably  of  a 
velocity  of  several  thousand  feet  per  second).  If  the  negative 
terminal  is  fluid,  the  cathode  spot  causes  a  depression,  by  the 
reaction  of  the  vapor  blast,  and  is  in  a  more  or  less  rapid  motion, 
depending  on  the  fluidity. 

As  the  arc  conductor  is  a  vapor  stream  of  electrode  material, 
this  vapor  stream  must  first  be  produced,  that  is,  energy  must  be 
expended  before  arc  conduction  can  take  place.  The  arc,  there- 
fore, does  not  start  spontaneously  between  the  arc  terminals,  if 
sufficient  voltage  is  supplied  to  maintain  the  arc  (as  is  the  case 
with  spark  conduction)  but  the  arc  has  first  to  be  started,  that 
is,  the  conducting  vapor  bridge  be  produced.  This  can  be  done 
by  bringing  the  electrodes  into  contact  and  separating  them,  or 
by  a  high-voltage  spark  or  Geissler  discharge,  or  by  the  vapor 
stream  of  another  arc,  or  by  producing  electronic  conduction,  as 
by  an  incandescent  filament.  Inversely,  if  the  current  in  the  arc 


32  ELECTRIC  CIRCUITS 

stopped  even  for  a  moment,  conduction  ceases,  that  is,  the  arc 
extinguishes  and  has  to  be  restarted.  Thus,  arc  conduction  may 
also  be  called  continuous  conduction. 

21.  The  arc  stream  is  conducting  only  in  the  direction  of  its 
motion,  but  not  in  the  reverse  direction.  Any  body,  which  is 
reached  by  the  arc  stream,  is  conductively  connected  with  it,  if 
positive  toward  it,  but  is  not  in  conductive  connection,  if  negative 
or  isolated,  since,  if  this  body  is  negative  to  the  arc  stream,  an  arc 
stream  would  have  to  issue  from  this  body,  to  connect  it  con- 
ductively, and  this  would  require  energy  to  be  expended  on  the 
body,  before  current  flows  to  it.  Thus,  only  if  the  arc  stream  is 
very  hot,  and  the  negative  voltage  of  the  body  impinged  by  it 
very  high,  and  the  body  small  enough  to  be  heated  to  high  tem- 
perature, an  arc  spot  may  form  on  it  by  heat  energy.  If,  there- 
fore, a  body  touched  by  the  arc  stream  is  connected  to  an  alternat- 
ing voltage,  so  that  it  is  alternately  positive  and  negative  toward 
the  arc  stream,  then  conduction  occurs  during  the  half-wave, 
when  this  body  is  positive,  but  no  conduction  during  the  negative 
half-wave  (except  when  the  negative  voltage  is  so  high  as  to  give 
disruptive  conduction),  and  the  arc  thus  rectifies  the  alternating 
voltage,  that  is,  permits  current  to  pass  in  one  direction  only. 
The  arc  thus  is  a  unidirectional  conductor,  and  as  such  extensively 
used  for  rectification  of  alternating  voltages.  Usually  vacuum 
arcs  are  employed  for  this  purpose,  mainly  the  mercury  arc,  due 
to  its  very  great  rectifying  range  of  voltage. 

Since  the  arc  is  a  unidirectional  conductor,  it  usually  can  not 
exist  with  alternating  currents  of  moderate  voltage,  as  at  the  end 
of  every  half-wave  the  arc  extinguishes.  To  maintain  an  alterna- 
ting arc  between  two  terminals,  a  voltage  is  required  sufficiently 
high  to  restart  the  arc  at  every  half -wave  by  jumping  an  elec- 
trostatic spark  between  the  terminals  through  the  hot  residual 
vapor  of  the  preceding  half-wave.  The  temperature  of  this  vapor 
is  that  of  the  boiling  point  of  the  electrode  material.  The  voltage 
required  by  the  electrostatic  spark,  that  is,  by  disruptive  conduc- 
tion, decreases  with  increase  of  temperature,  for  a  13-mm.  gap 
about  as  shown  by  curve  I  in  Fig.  18.  The  voltage  required  to 
maintain  an  arc,  that  is,  the  direct-current  voltage,  increases  with 
increasing  arc  temperature,  and  therefore  increasing  radiation, 
etc.,  about  as  shown  by  curve  II  in  Fig.  18.  As  seen,  the  curves 
I  and  II  intersect  at  some  very  high  temperature,  and  materials 
as  carbon,  which  have  a  boiling  point  above  this  temperature, 


ELECTRIC  CONDUCTION 


33 


require  a  lower  voltage  for  restarting  than  for  maintaining  the 
arc,  that  is,  the  voltage  required  to  maintain  the  arc  restarts  it 
at  every  half-wave  of  alternating  current,  and  such  materials  thus 
give  a  steady  alternating  arc.  Even  materials  of  a  somewhat 
lower  boiling  point,  in  which  the  starting  voltage  is  not  much 
above  the  running  voltage  of  the  arc,  maintain  a  steady  alter- 
nating arc,  as  in  starting  the  voltage  consumed  by  the  steadying 
resistance  or  reactance  is  available.  Electrode  materials  of  low 


FIG.  18. 

boiling  point,  however,  can  not  maintain  steady  alternating  arcs 
at  moderate  voltage. 

The  range  in  Fig.  18,  above  the  curve  I,  thus  is  that  in  which 
alternating  arcs  can  exist;  in  the  range  between  I  and  II,  an  alter- 
nating voltage  can  not  maintain  the  arc,  but  unidirectional  cur- 
rent is  produced  from  an  alternating  voltage,  if  the  arc  conductor 
is  maintained  by  excitation  of  its  negative  terminals,  as  by  an 
auxiliary  arc.  This,  therefore,  is  the  rectifying  range  of  arc  con- 
duction. Below  curve  II  any  conduction  ceases,  as  the  voltage  is 
insufficient  to  maintain  the  conducting  vapor  stream. 

Fig.  18  is  only  approximate.     As  ordinates  are  used  the  loga- 


34 


ELECTRIC  CIRCUITS 


rithm  of  the  voltage,  to  give  better  proportions.  The  boiling 
points  of  some  materials  are  approximately  indicated  on  the 
curves. 

It  is  essential  for  the  electrical  engineer  to  thoroughly  under- 
stand the  nature  of  the  arc,  not  only  because  of  its  use  as  illumi- 
nant,  in  arc  lighting,  but  more  still  because  accidental  arcs  are 
the  foremost  cause  of  instability  and  troubles  from  dangerous 
transients  in  electric  circuits. 


FIG.  19. 

22.  The  voltage  consumed  by  an  arc  stream,  e\t  at  constant 
current,  i,  is  approximately  proportional  to  the  arc  length,  I,  or 
rather  to  the  arc  length  plus  a  small  quantity,  d,  which  probably 
represents  the  cooling  effect  of  the  electrodes. 

Plotting  the  arc  voltage,  e,  as  function  of  the  current,  i,  at  con- 
stant arc  length,  gives  dropping  volt-ampere  characteristics,  and 
the  voltage  increases  with  decreasing  current  the  more,  the  longer 


ELECTRIC  CONDUCTION  35 

the  arc.     Such  characteristics  are  shown  in  Fig.  19  for  the  mag- 
netite arcs  of  0.3;  1.25;  2.5  and  3.75  cm.  length. 

These  curves  can  be  represented  with  good  approximation  by 
the  equation 

.    c(l  +  5) 

e  =  a  + /=—  (4) 

\A 

This  equation,  which  originally  was  derived  empirically,  can 
also  be  derived  by  theoretical  reasoning: 

Assuming  the  amount  of  arc  vapor,  that  is,  the  section  of  the 
conducting  vapor  stream,  as  proportional  to  the  current,  and  the 
heat  produced  at  the  positive  terminal  as  proportional  to  the 
vapor  stream  and  thus  the  current,  the  power  consumed  at  the 
terminals  is  proportional  to  the  current.  As  the  power  equals 
the  current  times  the  terminal  drop  of  voltage,  it  follows  that  this 
terminal  drop,  a,  is  constant  and  independent  of  current  or  arc 
length — similar  as  the  terminal  drop  at  the  electrodes  in  electro- 
lytic conduction  is  independent  of  the  current. 

The  power  consumed  in  the  arc  stream,  p\  =  eii,  is  given  off 
from  the  surface  of  the  stream,  by  radiation,  conduction  and  con- 
vection of  heat.  The  temperature  of  the  arc  stream  is  constant, 
as  that  of  the  boiling  point  of  the  electrode  material.  The  power, 
therefore,  is  proportional  to  the  surface  of  the  arc  stream,  that 
is,  proportional  to  the  square  root  of  its  section,  and  therefore 
the  square  root  of  the  current,  and  proportional  to  the  arc  length, 
/,  plus  a  small  quantity,  5,  which  corrects  for  the  cooling  effect 
of  the  electrodes.  This  gives 

Pi  =  ei  i  =  c  \/i  (I  H-  6) 
or, 

cd  +  S)  , 

ei=  ~^/r 

as  the  voltage  consumed  in  the  arc  stream. 

Since  a  represents  the  coefficient  of  power  consumed  in  produc- 
ing the  vapor  stream  and  heating  the  positive  terminal,  and  c  the 
coefficient  of  power  dissipated  from  the  vapor  stream,  a  and  c 
are  different  for  different  materials,  and  in  general  higher  for 
materials  of  higher  boiling  point  and  thus  higher  arc  tempera- 
ture, c,  however,  depends  greatly  on  the  gas  pressure  in  the 
space  in  which  the  arc  occurs,  and  decreases  with  decreasing  gas 
pressure.  It  is,  approximately,  when  I  is  given  in  centimeter  at 
atmospheric  pressure, 


36  ELECTRIC  CIRCUITS 

a  =  13   volts  for   mercury, 

=  16  volts  for  zinc  and  cadmium  (approximately), 

=  30  volts  for  magnetite, 

=  36  volts  for  carbon; 
c  =  31  for  magnetite, 

=  35  for  carbon; 
d  =  0.125  cm.  for  magnetite, 

=  0.8  cm.  for  carbon. 

The  least  agreement  with  the  equation  (4)  is  shown  by  the  car- 
bon arc.  It  agrees  fairly  well  for  arc  lengths  above  0.75  cm.,  but 
for  shorter  arc  lengths,  the  observed  voltage  is  lower  than  given 
by  equation  (4),  and  approaches  for  I  =  0  the  value  e  =  28  volts. 

It  seems  as  if  the  terminal  drop,  a  =  36  volts  with  carbon,  con- 
sists of  an  actual  terminal  drop,  a0  =  28  volts,  and  a  terminal 
drop  of  ai  =  8  volts,  which  resides  in  the  space  within  a  short 
distance  from  the  terminals. 

Stability  Curves  of  the  Arc 

23.  As  the  volt-ampere  characteristics  of  the  arc  show  a  de- 
crease of  voltage  with  increase  of  current,  over  the  entire  range  of 
current,  the  arc  is  unstable  on  constant  voltage  supplied  to  its 
terminals,  at  every  current. 

Inserting  in  series  to  a  magnetite  arc  of  1.8  cm.  length,  shown  as 
curve  I  in  Fig.  20,  a  constant  resistance  of  r  =  10  ohms,  the  vol- 
tage consumed  by  this  resistance  is  proportional  to  the  current, 
and  thus  given  by  the  straight  line  II  in  Fig.  20.  Adding  this 
voltage  II  to  the  arc-voltage  curve  I,  gives  the  total  voltage  con- 
sumed by  the  arc  and  its  series  resistance,  shown  as  curve  III. 
In  curve  III,  the  voltage  decreases  with  increase  of  current,  up  to 
io  =  2.9  amp.  and  the  arc  thus  is  unstable  for  currents  below 
2.9  amp.  For  currents  larger  than  2.9  amp.  the  voltage  increases 
with  increase  of  current,  and  the  arc  thus  is  stable.  The  point 
io  =  2.9  amp.  thus  separates  the  unstable  lower  part  of  curve 
III,  from  the  stable  upper  part. 

With  a  larger  series  resistance,  r'  =  20  ohms,  the  stability  range 
is  increased  down  to  1.7  amp.,  as  seen  from  curve  III,  but  higher 
voltages  are  required  for  the  operation  of  the  arc. 

With  a  smaller  series  resistance,  r"  =  5  ohms,  the  stability 
range  is  reduced  to  currents  above  4.8  amp.,  but  lower  voltages 
are  sufficient  for  the  operation  of  the  arc. 


ELECTRIC  CONDUCTION 


37 


At  the  stability  limit,  iQt  in  curve  III  of  Fig.  20,  the  resultant 

characteristic  is  horizontal,  that  is,  the  slope  of  the  resistance 

e' 
curve  II :  r  =  — »  is  equal  but  opposite  to  that  of  the  arc  charac- 


ii' 


ii 


in 


in 


in 


htf 

.130. 


FIG.  20. 

de 
teristic  I:  -7-.-     The  resistance,  r,  required  to  give  the  stability 

limit  at  current,  i,  thus  is  found  by  the  condition 

de 


Substituting  equation  (4)  into  (6)  gives 

+  5) 


r  = 


(6) 


(7) 


38  ELECTRIC  CIRCUITS 

as  the  minimum  resistance  to  produce  stability,  hence, 

n-.SS  +  IUia*,  (8) 

2\A' 

where  e\  =  arc  stream  voltage,  and 
E  =  e  +  ri 


is  the  minimum  voltage  required  by  arc  and  series  resistance, 
to  just  reach  stability. 

(9)  is  plotted  as  curve  IV  in  Fig.  20,  and  is  called  the  stability 
curve  of  the  arc.  It  is  of  the  same  form  as  the  arc  characteristic 
I,  and  derived  therefrom  by  adding  50  per  cent,  of  the  voltage, 
Ci,  consumed  by  the  arc  stream. 

The  stability  limit  of  an  arc,  on  constant  potential,  thus  lies 
at  an  excess  of  the  supply  voltage  over  the  arc  voltage  e  =  a  +  e\, 
by  50  per  cent,  of  the  voltage,  e\,  consumed  in  the  arc  stream. 
In  general,  to  get  reasonable  steadiness  and  absence  of  drifting 
of  current,  a  somewhat  higher  supply  voltage  and  larger  series 
resistance,  than  given  by  the  stability  curve  IV,  is  desirable. 

24.  The  preceding  applies  only  to  those  arcs  in  which  the  gas 
pressure  an  the  space  surrounding  the  arc,  and  thereby  the  arc 
vapor  pressure  and  temperature,  are  constant  and  independent 
of  the  current,  as  is  the  case  with  arcs  in  air,  at  "atmospheric 
pressure." 

With  arcs  in  which  the  vapor  pressure  and  temperature  vary 
with  the  current,  as  in  vacuum  arcs  like  the  mercury  arc,  different 
considerations  apply.  Thus,  in  a  mercury  arc  in  a  glass  tube, 
if  the  current  is  sufficiently  large  to  fill  the  entire  tube,  but  not 
so  large  that  condensation  of  the  mercury  vapor  can  not  freely 
occur  in  a  condensing  chamber,  the  power  dissipated  by  radiation, 
etc.,  may  be  assumed  as  proportional  to  the  length  of  the  tube, 
and  to  the  current 

p  =  e\i  =  di 
thus, 

<?i  =  d  (10) 

that  is,  the  stream  voltage  of  the  tube,  or  voltage  consumed  by 
the  arc  stream  (exclusive  terminal  drop)  is  independent  of  the 


ELECTRIC  CONDUCTION 


39 


current.     Adding  hereto  the  terminal  drop,  a,  gives  as  the  total 
voltage  consumed  by  the  mercury  tube 

e  =  a  +  cl  (11) 

for  a  mercury  arc  in  a  vacuum,  it  is  approximately 

c  =  ^  (12) 

where  d  =  diameter  of  the  tube,  since  the  diameter  of  the  tube 
is  proportional  to  the  surface  and  therefore  to  the  radiation 
coefficient. 
Thus, 

e  =  13  +  Ml  (13) 

At  high  currents,  the  vapor  pressure  rises  abnormally,  due  to 
incomplete  condensation,  and  the  voltage  therefore  rises,  and 


1 

VOLT-AMPERE  CH 
N 
L=40 

APPROX.      e 

ARACTERISTIC  OF  YACUU 
ERCURY  ARC 
CM.     D  =  2.2  CM. 
100 

M 

e. 

"    8.13  ~4.2t   -  5.6 

\ 

I 

45 

\ 

• 

40 

V 

^ 

*-«=; 

—    —  , 

—     — 

— 

—  .    —  ' 

;== 

—  —  = 

.—  —  - 

• 

.  ' 

***~~ 

_36 

so 

i 

—  > 

5 

! 

J 

1 

! 

' 

I 

) 

1 

0 

FIG.  21. 

at  low  currents  the  voltage  rises  again,  due  to  the  arc  not  filling 
the  entire  tube.  Such  a  volt-ampere  characteristic  is  given  in 
Fig.  21. 

25.  Herefrom  then  follows,  that  the  voltage  gradient  in  the 
mercury  arc,  for  a  tube  diameter  of  2  cm.,  is  about  %  volts  per 
centimeter  or  about  one-twentieth  of  what  it  is  in  the  Geissler 
tube,  and  the  specific  resistance  of  the  stream,  at  4  amp.,  is 


40  ELECTRIC  CIRCUITS 

about  0.2  ohms  per  cm.3,  or  of  the  magnitude  of  one  one- 
thousandth  of  what  it  is  in  the  Geissler  tube. 

At  higher  currents,  the  mercury  arc  in  a  vacuum  gives  a  rising 
volt-ampere  characteristic.  Nevertheless  it  is  not  stable  on 
constant-potential  supply,  as  the  rising  characteristic  applies  only 
to  stationary  conditions ;  the  instantaneous  characteristic  is  drop- 
ping. That  is,  if  the  current  is  suddenly  increased,  the  voltage 
drops,  regardless  of  the  current  value,  and  then  gradually,  with 
the  increasing  temperature  and  vapor  pressure,  increases  again,  to 
the  permanent  value,  a  lower  value  or  a  higher  value,  which- 
ever may  be  given  by  the  permanent  volt-ampere  characteristic. 

In  an  arc  at  atmospheric  pressure,  as  the  magnetite  arc,  the 
voltage  gradient  depends  on  the  current,  by  equation  (1),  and  at 
4  amp.  is  about  15  to  18  volts  per  centimeter.  The  specific  re- 
sistance of  the  arc  stream  is  of  the  magnitude  of  1  ohm  per  cm.3, 
and  less  with  larger  current  arcs,  thus  of  the  same  magnitude  as 
in  vacuum  arcs. 

Electronic   Conduction 

26.  Conduction  occurs  at  moderate  voltages  between  terminals 
in  a  partial  vacuum  as  well  as  in  a  perfect  vacuum,  if  the  terminals 
are  incandescent.     If  only  one  terminal  is  incandescent,  the  con- 
duction is  unidirectional,  that  is,  can  occur  only  in  that  direction, 
which  makes  the  incandescent  terminal  the  cathode,  or  negative. 
Such  a  vacuum  tube  then  rectifies  an  alternating  voltage  and  may 
be  used  as  rectifier.     If  a  perfect  vacuum  exists  in  the  conducting 
space  between  the  electrodes  of  such  a  hot  cathode  tube,  the  con- 
duction is  considered  as  true  electronic  conduction.     The  voltage 
consumed  by  the  tube  is  depending  on  the  high  temperature  of 
the  cathode,  and  is  of  the  magnitude  of  arc  voltages,  hence  very 
much  lower  than  in  the  Geissler  tube,  and  the  current  of  the  mag- 
nitude of  arc  currents,  hence  much  higher  than  in  the  Geissler  tube. 

27.  The  complete  volt-ampere  characteristic  of  gas  and  vapor 
conduction  thus  would  give  a  curve  of  the  shape  in  Fig.  22.     It 
consists  of  three  branches  separated  by  ranges  of  instability  or 
discontinuity.     The  branch  a,  at  very  low  current,  electronic  con- 
duction; the  branch  b,  discontinuous  or  Geissler  tube  conduction; 
and  the  branch  c,  arc  conduction.     The  change  from  a  to  b  oc- 
curs suddenly  and  abruptly,  accompanied  by  a  big  rise  of  current, 
as  soon  as  the  disruptive  voltage  is  reached.     The  change  b  to  c 


ELECTRIC  CONDUCTION 


41 


occurs  suddenly  and  abruptly,  by  the  formation  of  a  cathode  spot, 
anywhere  in  a  wide  range  of  current,  and  is  accompanied  by  a 
sudden  drop  of  voltage.  To  show  the  entire  range,  as  abscissae 
are  used  \/i  and  as  ordinates 


APPROXIMATE    VOLT    AMPERE 
CHARACTERISTIC  OF 
GASEOUS  CONDUCTION 


4000 


3000 


2000 


1000. 


500 


200 


FIG.  22. 


Review 

28.  The  various  classes  of  conduction:  metallic  conduction, 
electrolytic  conduction,  pyroelectric  conduction,  insulation,  gas 
vapor  and  electronic  conduction,  are  only  characteristic  types, 
but  numerous  intermediaries  exist,  and  transitions  from  one  type 
to  another  by  change  of  electrical  conditions,  of  temperature, 
etc. 

As  regards  to  the  magnitude  of  the  specific  resistance  or  resist- 
ivity, the  different  types  of  conductors  are  characterized  about  as 
follows : 


42  ELECTRIC  CIRCUITS 

The  resistivity  of  metallic  conductors  is  measured  in  microhm- 
centimeters. 

The  resistivity  of  electrolytic  conductors  is  measured  in  ohm- 
centimeters. 

The  resistivity  of  insulators  is  measured  in  megohm-centimeters 
and  millions  of  megohm-centimeters. 

The  resistivity  of  typical  pyroelectric  conductors  is  of  the  mag- 
nitude of  that  of  electrolytes,  ohm-centimeters,  but  extends  from 
this  down  toward  the  resistivities  of  metallic  conductors,  and  up 
toward  that  of  insulators. 

The  resistivity  of  gas  and  vapor  conduction  is  of  the  magnitude 
of  electrolytic  conduction:  arc  conduction  of  the  magnitude  of 
lower  resistance  electrolytes,  Geissler  tube  conduction  and  corona 
conduction  of  the  magnitude  of  higher-resistance  electrolytes. 

Electronic  conduction  at  atmospheric  temperature  is  of  the 
magnitude  of  that  of  insulators;  with  incandescent  terminals,  it 
reaches  the  magnitude  of  electrolytic  conduction. 

While  the  resistivities  of  pyroelectric  conductors  extend  over 
the  entire  range,  from  those  of  metals  to  those  of  insulators, 
typical  are  those  pyroelectric  conductors  having  a  resistivity  of 
electrolytic  conductors.  In  those  with  lower  resistivity,  the 
drop  of  the  volt-ampere  characteristic  decreases  and  the  insta- 
bility characteristic  becomes  less  pronounced;  in  those  of  higher 
resistivity,  the  negative  slope  becomes  steeper,  the  instability  in- 
creases, and  streak  conduction  or  finally  disruptive  conduction 
appears.  The  streak  conduction,  described  on  the  pyroelectric 
conductor,  probably  is  the  same  phenomenon  as  the  disruptive 
conduction  or  breakdown  of  insulators.  Just  as  streak  conduc- 
tion appears  most  under  sudden  application  of  voltage,  but  less 
under  gradual  voltage  rise  and  thus  gradual  heating,  so  insulators 
of  high  disruptive  strength,  when  of  low  resistivity  by  absorbed 
moisture,  etc.,  may  stand  indefinitely  voltages  applied  intermit- 
tently— so  as  to  allow  time  for  temperature  equalization — while 
quickly  breaking  down  under  very  much  lower  sustained  voltage. 


CHAPTER  III 

MAGNETISM 

Reluctivity 

29.  Considering  magnetism  as  the  phenomena  of  a  ' 'magnetic 
circuit/'  the  foremost  differences  between  the  characteristics 
of  the  magnetic  circuit  and  the  electric  circuit  are : 

(a)  The  maintenance  of  an  electric  circuit  requires  the  ex- 
penditure of  energy,  while  the  maintenance  of  a  magnetic  circuit 
does  not  require  the  expenditure  of  energy,  though  the  starting 
of  a  magnetic  circuit  requires  energy.  A  magnetic  circuit,  there- 
fore, can  remain  "remanent"  or  " permanent." 

(6)  All  materials  are  fairly  good  carriers  of  magnetic  flux, 
and  the  range  of  magnetic  permeabilities  is,  therefore,  narrow, 
from  1  to  a  few  thousands,  while  the  range  of  electric  conductivi- 
ties covers  a  range  of  1  to  1018.  The  magnetic  circuit  thus  is 
analogous  to  an  uninsulated  electric  circuit  immersed  in  a  fairly 
good  conductor,  as  salt  water:  the  current  or  flux  can  not  be 
carried  to  any  distance,  or  constrained  in  a  "conductor,"  but 
divides,  "leaks"  or  "strays." 

(c)  In  the  electric  circuit,  current  and  e.m.f.  are  proportional, 
in  most  cases;  that  is,  the  resistance  is  constant,  and  the  circuit 
therefore  can  be  calculated  theoretically.  In  the  magnetic 
circuit,  in  the  materials  of  high  permeability,  which  are  the  most 
important  carriers  of  the  magnetic  flux,  the  relation  between  flux, 
m.m.f.  and  energy  is  merely  empirical,  the  "reluctance"  or  mag- 
netic resistance  is  not  constant,  but  varies  with  the  flux  density, 
the  previous  history,  etc.  In  the  absence  of  rational  laws,  most 
of  the  magnetic  calculations  thus  have  to  be  made  by  taking 
numerical  values  from  curves  or  tables. 

The  only  rational  law  of  magnetic  relation,  which  has  not  been 
disproven,  is  Frohlich's  (1882) : 

11  The  premeability  is  proportional  to  the  magnetizability" 

»  =  a(S-  B)  (1) 

where  B  is  the  magnetic  flux  density,  S  the  saturation  density, 

43 


44  ELECTRIC  CIRCUITS 

and  S  —  B  therefore  the  magnetizability,  that  is,  the  still  avail- 
able increase  of  flux  density,  over  that  existing. 
From  (1)  follows,  by  substituting, 

*- 

and  rearranging, 


*- 


B  - 


where 

<r  =  -«  =  saturation  coefficient,  that   is,  the  reciprocal  of  the 
saturation  value,  S,  of  flux  density,  B,  and 


= 


for  B  =  0,  equation  (1)  gives 


ju0  =  a*S  =  -;«  =  -  (4) 


that  is,  a  is  the  reciprocal  of  the  magnetic  permeability  at  zero 
flux  density. 

A  very  convenient  form  of  this  law  has  been  found  by  Kennelly 
(1893)  by  introducing  the  reciprocal  of  the  permeability,  as 
reluctivity  p, 

1       H 
p  =  M  =  B  ' 

in  the  form,  which  can  be  derived  from  (3)  by  transposition. 

p  =  a+(rH  (5) 

As  a  dominates  the  reluctivity  at  lower  magnetizing  forces, 
and  thereby  the  initial  rate  of  rise  of  the  magnetization  curve, 
which  is  characteristic  of  the  "magnetic  hardness"  of  the  material, 
it  is  called  the  coefficient  of  magnetic  hardness. 

30.  When  investigating  flux  densities,  B,  at  very  high  field 
intensities,  H,  it  was  found  that  B  does  not  reach  a  finite  satura- 
tion value,  but  increases  indefinitely;  that,  however, 

Bo  =  B-H  (6) 

reaches  a  finite  saturation  value  S,  which  with  iron  usually  is  not 
far  from  20  kilolines  per  cm.2,  and  that  therefore  Frohlich's  and 
Kennelly's  laws  apply  not  to  B,  but  to  BQ.  The  latter,  then, 


MAGNETISM 


45 


lie 


is  usually  called  the  metallic  magnetic  density  or  ferromagnet 
density. 

Bo  may  be  considered  as  the  magnetic  flux  carried  by  the  mole- 
cules of  the  iron  or  other  magnetic  material,  in  addition  to  the 


FIG.  23. 

space  flux,  H,  or  flux  carried  by  space  independent  of  the  material 
in  space. 

The  best  evidence  seems  to  corroborate,  that  with  the  excep- 
tion of  very  low  field  intensities  (where  the  customary  magneti- 
zation curve  usually  has  an  inward  bend,  which  will  be  discussed 
later)  in  perfectly  pure  magnetic  materials,  iron,  nickel,  cobalt, 


46  ELECTRIC  CIRCUITS 

etc.,  the  linear  law  of  reluctivity  (5)  and  (3)  is  rigidly  obeyed  by 
the  metallic  induction  B0. 

In  the  more  or  less  impure  commercial  materials,  however,  the 
p  —  H  relation,  while  a  straight  line,  often  has  one,  and  occasion- 
ally two  points,  where  its  slope,  and  thus  the  values  of  a  and  a 
change. 

Fig.  23  shows  an  average  magnetization  curve,  of  good  standard 
iron,  with  field  intensity,  H,  as  abscissae,  and  magnetic  induction, 
B,  as  ordinates.  The  total  induction  is  shown  in  drawn  lines,  the 
metallic  induction  in  dotted  lines.  The  ordinates  are  given  in 
kilolines  per  cm.2,  the  abscissa  in  units  for  B\.,  in  tens  for  #2,  and 
in  hundreds  for  #3. 

The  reluctivity  curves,  for  the  three  scales  of  abscissae,  are 
plotted  as  pi,  p2,  PS,  in  tenths  of  milli-units,  in  milli-units  and  in 
tens  of  milli-units. 

Below  H  =  3,  p  is  not  a  straight  line,  but  curved,  due  to  the  in- 
ward bend  of  the  magnetization  curve,  B,  in  this  range.  The 
straight-line  law  is  reached  at  the  point  d,  at  H  =  3,  and  the  re- 
luctivity is  then  expressed  by  the  linear  law 

Pl  =  0.102  +  0.059  H  (7) 

for 

3  <  H  <  18, 

giving  an  apparent  saturation  value, 

51  =  16,950. 

At  H  =  18,  a  bend  occurs  in  the  reluctivity  line,  marked  by 
point  02,  and  above  this  point  the  reluctivity  follows  the  equation 

P2  =  0.18  +  0.0548  H  (8) 

for 

18  <  H  <  80, 

giving  an  apparent  saturation  value 

52  =  18,250. 

At  H  =  80,  another  bend  occurs  in  the  reluctivity  line,  marked 
by  point  c3,  and  above  this  point,  up  to  saturation,  the  reluctivity 
follows  the  equation 

p3  =  0.70  +  0.0477  H  (9) 

for 

#>80 

giving  the  true  saturation  value, 

S  =  20,960. 


MAGNETISM 


47 


Point  c2  is  frequently  absent. 

Fig.  24  gives  once  more  the  magnetization  curve  (metallic  in- 
duction) as  B,  and  gives  as  dotted  curves  BI,  B2  and  B3  the  mag- 
netization curves  calculated  from  the  three  linear  reluctivity  equa- 
tions (7),  (8),  (9).  As  seen,  neither  of  the  equations  represents 


FIG.  24. 


B  even  approximately  over  the  entire  range,  but  each  represents 
it  very  accurately  within  its  range.  The  first,  equation  (7) ,  prob- 
ably covers  practically  the  entire  industrially  important  range. 

37.  As  these  critical  points  c2  and  c3  do  not  seem  to  exist  in  per- 
fectly pure  materials,  and  as  the  change  of  direction  of  the  re- 


48  ELECTRIC  CIRCUITS 

luctivity  line  is  in  general  the  greater,  the  more  impure  the  mate- 
rial, the  cause  seems  to  be  lack  of  homogeneity  of  the  material; 
that  is,  the  presence,  either  on  the  surface  as  scale,  or  in  the  body, 
as  inglomerate,  of  materials  of  different  magnetic  characteristics: 
magnetite,  cementite,  silicide.  Such  materials  have  a  much 
greater  hardness,  that  is,  higher  value  of  a,  and  thereby  would 
give  the  observed  effect.  At  low  field  intensities,  H,  the  harder 
material  carries  practically  no  flux,  and  all  the  flux  is  carried  by 
the  soft  material.  The  flux  density  therefore  rises  rapidly,  giving 
low  «,  but  tends  toward  an  apparent  low  saturation  value,  as 
the  flux-carrying  material  fills  only  part  of  the  space.  At  higher 
field  intensities,  the  harder  material  begins  to  carry  flux,  and 
while  in  the  softer  material  the  flux  increases  less,  the  increase  of 
flux  in  the  harder  material  gives  a  greater  increase  of  total  flux 
density  and  a  greater  saturation  value,  but  also  a  greater  hard- 
ness, as  the  resultant  of  both  materials. 

Thus,  if  the  magnetic  material  is  a  conglomerate  of  fraction  p 
of  soft  material  of  reluctivity  p\  (ferrite)  and  q  =  1  —  p  of  hard 
material  of  reluctivity,  p2  (cementite,  silicide,  magnetite), 

Pi  =  «i  +  viH  \ 

P2 

at  low  values  of  H,  the  part  p  of  the  section  carries  flux  by  pi,  the 
part  q  carries  flux  by  p2,  but  as  p2  is  very  high  compared  with  pi, 
the  latter  flux  is  negligible,  and  it  is 


rr  1  (10) 

=    Oiz   +  0-2/2    I 


+       H  (H) 

p        p        p 

At  high  values  of  H,  the  flux  goes  through  both  materials,  more  or 
less  in  series,  and  it  thus  is 

p"  =  ppi  +  qp2  =  (pen  +  qaz)  +  (p<ri  +  qa2)H          (12) 

if  we  assume  the  same  saturation  value,  <r,  for  both  materials,  and 
neglect  a\  compared  with  «2,  it  is 

p"  =  q  az  +  ffH  (13) 

Substituting,   as  instance,    (7)   and    (9)   into   (11)   and   (13) 
respectively,  gives 

2!  =  0.102, 

-  =  0.059, 
P 


MAGNETISM  49 

qa2  =  0.70, 

a  =  0.0477, 
hence 

p  =  0.80  :     Pl  =  0.082  +  0.0477  H, 
q  =  0.20  :     p2  =      3.5  +  0.0477  H. 

However,  the  saturation  coefficients,  o-,  of  the  two  materials 
probably  are  usually  not  equal. 

The  deviation  of  the  reluctivity  equation  from  a  straight  line, 
by  the  change  of  slope  at  the  critical  points,  c2  and  c3,  thus  probably 
is  only  apparent,  and  is  the  outward  appearance  of  a  change  of 
the  flux  carrier  in  an  unhomogeneous  material,  that  is,  the  result 
of  a  second  and  magnetically  harder  material  beginning  to  carry 
flux. 

Such  bends  in  the  reluctivity  line  have  been  artificially  produced 
by  Mr.  John  D.  Ball  in  combining  by  superposition  two  different 
materials,  which  separately  gave  straight-line,  p,  curves,  while 
combined  they  gave  a  curve  showing  the  characteristic  bend. 

Very  impure  materials,  like  cast  iron,  may  give  throughout  a 
curved  reluctivity  line. 

32.  For  very  low  values  of  field  intensity,  H  <  3,  however,  the 
straight-line  law  of  reluctivity  apparently  fails,  and  the  mag- 
netization curve  in  Fig.  23  has  an  inward  bend,  which  gives  rise 
of  p  with  decreasing  H. 

This  curve  is  taken  by  ballistic  galvanometer,  by  the  step-by- 
step  method,  that  is,  H  is  increased  in  successive  steps,  and  the 
increase  of  B  observed  by  the  throw  of  the  galvanometer  needle. 
It  thus  is  a  "rising  magnetization  curve." 

The  first  part  of  this  curve  is  in  Fig.  25  reproduced,  as  B\, 
in  twice  the  abscissae  and  half  the  ordinates,  so  as  to  give  it  an 
average  slope  of  45°,  as  with  this  slope  curve  shapes  such  as  the 
inward  bend  of  BI  below  H  =  2,  are  best  shown  ("Engineering 
Mathematics,"  p.  286). 

Suppose  now,  at  some  point,  B0  =  13.15,  we  stop  the  increase 
of  H,  and  decrease  again,  down  to  0.  We  do  not  return  on  the 
same  magnetization  curve,  B\,  but  on  another  curve,  B'i,  the 
"decreasing  magnetic  characteristic,"  and  at  H  =  0,  we  are  not 
back  to  B  =  0,  but  a  residual  or  remanent  flux  is  left,  in  Fig.  25 : 
R  =  7.4. 

Where  the  magnetic  circuit  contains  an  air-gap,  as  the  field 
circuits  of  electrical  machinery,  the  decreasing  magnetic  charac- 
teristic, B'i,  is  very  much  nearer  to  the  increasing  one,  BI,  than  in 


50  ELECTRIC  CIRCUITS 

the  closed  magnetic  circuit,  Fig.  25,  and  practically  coincides  for 
higher  values  of  H. 

There  appears  no  theoretical  reason  why  the  rising  character- 
istic, BI,  should  be  selected  as  the  representative  magnetization 
curve,  and  not  the  decreasing  characteristic,  B'i,  except  the  inci- 
dent, that  BI  passes  through  zero.  In  many  engineering  applica- 
tions, for  instance,  the  calculation  of  the  regulation  of  a  generator, 
that  is,  the  decrease  of  voltage  under  increase  of  load,  it  is  ob- 
viously the  decreasing  characteristic,  B'i,  which  is  determining. 

Suppose  we  continue  B\  into  negative  values  of  H,  to  the  point 
A i,  at  H  —  —1.5,  B  =  —4,  and  then  again  reverse,  we  get  a  ris- 
ing magnetization  curve,  B",  which  passes  H  =  0  at  a  negative 
remanent  magnetism.  Suppose  we  stop  at  point  A2,  at  H  = 
—  1.12,  B  =  —1.0:  the  rising  magnetization  curve  B'"  then  passes 
H  =  0  at  a  positive  remanent  magnetism.  There  must  thus  be 
a  point,  AQ,  between  AI  and  A2,  such  that  the  rising  magnetiza- 
tion curve,  B',  starting  from  A0,  passes  through  the  zero  point 
H  =  0,  B  =  0,  and  thereby  runs  into  the  curve,  BI. 

The  rising  magnetization  curve,  or  standard  magnetic  charac- 
teristic determined  by  the  step-by-step  method,  BI,  thus  is  noth- 
ing but  the  rising  branch  of  an  unsymmetrical  hysteresis  cycle, 
traversed  between  such  limits  +BQ  and  —  A0,  that  the  rising 
branch  of  the  hysteresis  cycle  passes  through  the  zero  point. 

33.  The  characteristic  shape  of  a  hysteresis  cycle  is  that  it  is  a 
loop,  pointed  at  either  end  and  thereby  having  an  inflexion  point 
about  the  middle  of  either  branch.  In  the  unsymmetrical  loop 
+Bi,  —AQ  of  Fig.  25,  the  zero  point  is  fairly  close  to  one  extreme, 
AQ,  and  the  inflexion  point,  characteristic  of  the  hysteresis  loop, 
thus  lies  between  0  and  B0,  that  is,  on  that  part  of  the  rising 
branch,  which  is  used  as  the  "magnetic  characteristic,"  BI, 
and  thereby  produces  the  inward  bend  in  the  magnetization  curve 
at  low  fields,  which  has  always  been  so  puzzling. 

If,  however,  we  would  stop  the  increase  of  H  at  B"0,  we  would 
get  the  decreasing  magnetization  curve,  B"i,  and  still  other 
curves  for  other  starting  points  of  the  decreasing  characteristic. 

Thus,  the  relation  between  magnetic  flux  density,  B,  and  mag- 
metic  field  intensity,  H,  is  not  definite,  but  any  point  between  the 
various  rising  and  decreasing  characteristics  B",  BI,  B"',  E'\, 
B'i,  and  for  some  distance  outside  thereof,  is  a  possible  B-H 
relation.  BI  has  the  characteristic  that  it  passes  through  the 
zero  point.  But  it  is  not  the  only  characteristic  which  does  this : 


MAGNETISM 


51 


if  we  traverse  the  hysteresis  cycle  between  the  unsymmetrical 
limits  +A0  and  —  BQ,  as  shown  in  Fig.  26,  its  decreasing  branch 
B 3  passes  through  the  zero  point,  that  is,  has  the  same  feature 
as  BI.  It  is  interesting  to  note,  that  Bs  does  not  show  an  inward 
bend,  and  the  reluctivity  curve  of  B3,  given  as  p$  in  Fig.  28, 
apparently  is  a  straight  line. 

Magnetic  characteristics  are  frequently  determined  by  the 
method  of  reversals,  by  reversing  the  field  intensity,  H,  and  ob- 
serving the  voltage  induced  thereby  by  ballistic  galvanometer, 


FIGS.  25  AND  26. 

or  using  an  alternating  current  for  field  excitation,  and  observing 
the  induced  alternating  voltage,  preferably  by  oscillograph  to 
eliminate  wave-shape  error. 

This  "alternating  magnetic  characteristic"  is  the  one  which  is 
of  consequence  in  the  design  of  alternating-current  apparatus. 
It  differs  from  the  " rising  magnetic  characteristic,"  BI  by  giving 
lower  values  of  B,  for  the  same  H,  materially  so  at  low  values  of  H. 
It  shows  the  inward  bend  at  low  fields  still  more  pronounced  than 
BI  does.  It  is  shown  as  curve  B2  in  Fig.  27,  and  its  reluctivity 


52 


ELECTRIC  CIRCUITS 


line  given  as  p2  in  Fig.  28.  At  higher  values  of  H:  from  H  =  3  up- 
ward, BI  and  B2  both  coincide  with  the  curve,  B0,  representing  the 
straight-line  reluctivity  law. 


14- 


4 


10 


V 


-3 


-1 


L- 


A 

-12 


-14 


FIG.  27. 


-3 
10- 


7o 


FIG.  28. 


The  alternating  characteristic,  B2,  is  not  a  branch  of  any  hystere- 
sis cycle.  It  is  reproducible  and  independent  of  the  previous 
history  of  the  magnetic  circuit,  except  perhaps  at  extremely  low 
values  of  H,  and  in  view  of  its  engineering  importance  as  repre- 


MAGNETISM  53 

senting  the  conditions  in  the  alternating  magnetic  field,  it  would 
appear  the  most  representative  magnetic  characteristic,  and  is 
commonly  used  as  such. 

It  has,  however,  the  disadvantage  that  it  represents  an  un- 
stable condition. 

Thus  in  Fig.  27,  an  alternating  field  H  =  I  gives  an  alternating 
flux  density,  B2  =  2.6.  If,  however,  this  field  strength  H  =  1 
is  left  on  the  magnetic  circuit,  the  flux  does  not  remain  at  B2  = 
2.6,  but  gradually  creeps  up  to  higher  values,  especially  in  the 
presence  of  mechanical  vibrations  or  slight  pulsations  of  the 
magnetizing  current.  To  a  lesser  extent,  the  same  occurs  with 
the  values  of  curve,  B\,  to  a  greater  extent  with  J53.  At  very  low 
densities,  this  creepage  due  to  instability  of  the  B-H  relation  may 
amount  to  hundreds  of  per  cent,  and  continue  to  an  appreciable 
extent  for  minutes,  and  with  magnetically  hard  materials  for 
many  years.  Thus  steel  structures  in  the  terrestrial  magnetic 
field  show  immediately  after  erection  only  a  small  part  of  the 
magnetization,  which  they  finally  assume,  after  many  years. 

Thus  the  alternating  characteristic,  B2,  however  important  in 
electrical  engineering,  can,  due  to  its  instability,  not  be  considered 
as  representing  the  true  physical  relation  between  B  and  H  any 
more  than  the  branches  of  hysteresis  cycles  BI  and  Bz. 

34.  Correctly,  the  relation  between  B  and  H  thus  can  not  be 
expressed  by  a  curve,  but  by  an  area. 

Suppose  a  hysteresis  cycle  is  performed  between  infinite  values 
of  field  intensity:  H  =  ±  oc ,  that  is,  practically,  between  very 
high  values  such  as  are  given  for  instance  by  the  isthmus  method 
of  magnetic  testing  (where  values  of  H  of  over  40,000  have  been 
reached.  Very  much  lower  values  probably  give  practically  the 
same  curve).  This  gives  a  magnetic  cycle  shown  in  Fig.  5  as 
B',  B".  Any  point,  H,  B,  within  the  area  of  this  loop  between  B' 
and  B"  of  Fig.  27  then  represents  a  possible  condition  of  the 
magnetic  circuit,  and  can  be  reached  by  starting  from  any  other 
point,  HQ,  BQ,  such  as  the  zero  point,  by  gradual  change  of  H . 

Thus,  for  instance,  from  point  P0,  the  points  PI,  P2,  PS,  etc.,  are 
reached  on  the  curves  shown  in  the  dotted  lines  in  Fig.  27. 

As  seen  from  Fig.  27,  a  given  value  of  field  intensity,  such  as 
H  =  1,  may  give  any  value  of  flux  density  between  B  —  —4.6 
and  B  =  +13.6,  and  a  given  value  of  flux  density,  such  as  B  = 
10,  may  result  from  any  value  of  field  intensity,  between  H  = 
-  0.25  to  H  =  +  3.4 


54  ELECTRIC  CIRCUITS 

The  different  values  of  J5,  corresponding  to  the  same  value  of  H 
in  the  magnetic  area,  Fig.  27,  are  not  equally  stable,  but  the  val- 
ues near  the  limits  B'  and  B"  are  very  unstable,  and  become  more 
stable  toward  the  interior  of  the  area.  Thus,  the  relation  of 
point  Pi,  Fig.  27:  H  =  2,  B  =  13,  would  rapidly  change,  by  the 
flux  density  decreasing,  to  P0,  slower  to  P2  and  then  still  slower, 
while  from  point  P3  the  flux  density  would  gradually  creep  up. 

If  thus  follows,  that  somewhere  between  the  extremes  B'  and 
B",  which  are  most  unstable,  there  must  be  a  value  of  B}  which  is 
stable,  that  is,  represents  the  stationary  and  permanent  relation 
between  B  and  H,  and  toward  this  stable  value,  J50,  all  other  val- 
ues would  gradually  approach.  This,  then,  would  give  the  true 
magnetic  characteristic:  the  stable  physical  relation  between  B 
and#. 

At  higher  field  intensities,  beyond  the  first  critical  point,  Ci, 
this  stable  condition  is  rapidly  reached,  and  therefore  is  given  by 
all  the  methods  of  determining  magnetic  characteristics.  Hence, 
the  curves  BI,  Bz,  BQ  coincide  there,  and  the  linear  law  of  re- 
luctivity applies.  Below  Ci,  however,  the  range  of  possible,  B, 
values  is  so  large,  and  the  final  approach  to  the  stable  value  so 
slow,  as  to  make  it  difficult  of  determination. 

36.  For  H  =  0,  the  magnetic  range  is  from  —  R0  =  —11.2  to 
+Ro  =  11.2;  the  permanent  value  is  zero.  The  method  of  reach- 
ing the  permanent  value,  whatever  may  be  the  remanent  mag- 
netism, is  well  known;  it  is  by  ''demagnetizing"  that  is,  placing 
the  material  into  a  powerful  alternating  field,  a  demagnetizing 
coil,  and  gradually  reducing  this  field  to  zero.  That  is,  describ- 
ing a  large  number  of  cycles  with  gradually  decreasing  amplitude. 

The  same  can  be  applied  to  any  other  point  of  the  magnetiza- 
tion curve.  Thus  f or  H  =  1,  to  reach  permanent  condition,  an 
alternating  m.m.f.  is  superimposed  upon  H  =  1,  and  gradually 
decreased  to  zero,  and  during  these  successive  cycles  of  decreas- 
ing amplitude,  with  H  =  1,  as  mean  value,  the  flux  density  gradu- 
ally approaches  its  permanent  or  stable  value.  (The  only  re- 
quirement is,  that  the  initial  alternating  field  must  be  higher  than 
any  unidirectional  field  to  which  the  magnetic  circuit  had  been 
exposed.) 

This  seems  to  be  the  value  given  by  curve  BQ,  that  is,  by  the 
straight-line  law  of  reluctivity.  In  other  words,  it  is  probable 
that: 

Frohlich's  equation,  or  Kennelly's  linear  law  of  reluctivity 


MAGNETISM  55 

represent  the  permanent  or  stable  relation  between  B  and  H, 
that  is,  the  true  magnetic  characteristic  of  the  material,  over  the 
entire  range  down  to  H  =  0,  and  the  inward  bend  of  the  magnetic 
characteristic  for  low  field  intensities,  and  corresponding  increase 
of  reluctivity  p,  is  the  persistence  of  a  condition  of  magnetic 
instability,  just  as  remanent  and  permanent  magnetism  are. 

In  approaching  stable  conditions  by  the  superposition  of  an 
alternating  field,  this  field  can  be  applied  at  right  angles  to  the 
unidirectional  field,  as  by  passing  an  alternating  current  length- 
wise, that  is,  in  the  direction  of  the  lines  of  magnetic  force,  through 
the  material  of  the  magnetic  circuit.  This  superimposes  a  cir- 
cular alternating  flux  upon  the  continuous-length  flux,  and  per- 
mits observations  while  the  circular  alternating  flux  exists,  since 
the  latter  does  not  induce  in  the  exploring  circuit  of  the  former. 
Some  20  years  ago  Ewing  has  already  shown,  that  under  these 
conditions  the  hysteresis  loop  collapses,  the  inward  bend  of  the 
magnetic  characteristic  practically  vanishes,  and  the  magnetic 
characteristic  assumes  a  shape  like  curve  BQ. 

To  conclude,  then,  it  is  probable  that: 

In  pure  homogeneous  magnetic  materials,  the  stable  relation 
between  field  intensity,  H,  and  flux  density,  B,  is  expressed,  over 
the  entire  range  from  zero  to  infinity,  by  the  linear  equation 
of  reluctivity 

p  =  a  +  <rH, 

where   p   applies   to   the   metallic   magnetic  induction,  B  —  H. 

In  unhomogeneous  materials,  the  slope  of  the  reluctivity  line 
changes  at  one  or  more  critical  points,  at  which  the  flux  path 
changes,  by  a  material  of  greater  magnetic  hardness  beginning 
to  carry  flux. 

At  low  field  intensities,  the  range  of  unstable  values  of  B  is 
very  great,  and  the  approach  to  stability  so  slow,  that  considerable 
deviation  of  B  from  its  stable  value  can  persist,  sometimes  for 
years,  in  the  form  of  remanent  or  permanent  magnetism,  the 
inward  bend  of  the  magnetic  characteristic,  etc. 


CHAPTER  IV 

MAGNETISM 

Hysteresis 

36.  Unlike  the  electric  current,  which  requires  power  for  its 
maintenance,  the  maintenance  of  a  magnetic  flux  does  not  require 
energy  expenditure  (the  energy  consumed  by  the  magnetizing 
current  in  the  ohmic  resistance  of  the  magnetizing  winding  being 
an  electrical  and  not  a  magnetic  effect),  but  energy  is  required 
to  produce  a  magnetic  flux,  is  then  stored  as  potential  energy  in 
the  magnetic  flux,  and  is  returned  at  the  decrease  or  disappear- 
ance of  the  magnetic  flux.  However,  the  amount  of  energy  re- 
turned at  the  decrease  of  magnetic  flux  is  less  than  the  energy 
consumed  at  the  same  increase  of  magnetic  flux,  and  energy  is 
therefore  dissipated  by  the  magnetic  change,  by  conversion  into 
heat,  by  what  may  be  called  molecular  magnetic  friction,  at  least 
in  those  materials,  which  have  permeabilities  materially  higher 
than  unity. 

Thus,  if  a  magnetic  flux  is  periodically  changed,  between 
+  B  and  —  B,  or  between  BI  and  J52,  as  by  an  alternating  or  pul- 
sating current,  a  dissipation  of  energy  by  molecular  friction 
occurs  during  each  magnetic  cycle.  Experiment  shows  that  the 
energy  consumed  per  cycle  and  cm.3  of  magnetic  material  depends 
only  on  the  limits  of  the  cycle,  BI  and  B2,  but  not  on  the  speed  or 
wave  shape  of  the  change. 

If  the  energy  which  is  consumed  by  molecular  friction  is  sup- 
plied by  an  electric  current  as  magnetizing  force,  it  has  the  effect 
that  the  relations  between  the  magnetizing  current,  i,  or  magnetic 
field  intensity,  H,  and  the  magnetic  flux  density,  B,  is  not  revers- 
ible, but  for  rising,  H,  the  density,  B,  is  lower  than  for  decreasing 
H ;  that  is,  the  magnetism  lags  behind  the  magnetizing  force,  and 
the  phenomenon  thus  is  called  hysteresis,  and  gives  rise  to  the 
hysteresis  loop. 

However,  hysteresis  and  molecular  magnetic  friction  are  not 

56 


MAGNETISM  5? 

the  same  thing,  but  the  hysteresis  loop  is  the  measure  of  the  mo- 
lecular magnetic  friction  only  in  that  case,  when  energy  is  supplied 
to  or  abstracted  from  the  magnetic  circuit  only  by  the  magnetiz- 
ing current,  but  not  otherwise.  Thus,  if  mechanical  work  is  done 
by  the  magnetic  cycle — as  when  attracting  and  dropping  an  arma- 
ture— the  hysteresis  loops  enlarge,  representing  not  only  the 
energy  dissipated  by  molecular  magnetic  friction,  but  also  that 
converted  into  mechanical  work.  Inversely,  if  mechanical  en- 
ergy is  supplied  to  the  magnetic  circuit  as  by  vibrating  it  mechan- 
ically, the  hysteresis  loop  collapses  or  overturns,  and  its  area 
becomes  equal  to  the  molecular  magnetic  friction  minus  the 
mechanical  energy  absorbed.  The  reaction  machine,  as  synchron- 
ous motor  and  as  generator,  is  based  on  this  feature.  See 
"Reaction  Machine,"  "Theory  and  Calculation  of  Electrical 
Apparatus. " 

In  general,  when  speaking  of  hysteresis,  molecular  magnetic 
friction  is  meant,  and  the  hysteresis  cycle  assumed  under  the  con- 
dition of  no  other  energy  conversion,  and  this  assumption  will  be 
made  in  the  following,  except  where  expressly  stated  otherwise. 

The  hysteresis  cycle  is  independent  of  the  frequency  within 
commercial  frequencies  and  far  beyond  this  range.  Even  at 
frequencies  of  hundred  thousand  cycles,  experimental  evidence 
seems  to  show  that  the  hysteresis  cycle  is  not  materially  changed, 
except  in  so  far  as  eddy  currents  exert  a  demagnetizing  action  and 
thereby  require  a  change  of  the  impressed  m.m.f .,  to  get  the  same 
resultant  m.m.f.,  and  cause  a  change  of  the  magnetic  flux  dis- 
tribution by  their  screening  effect. 

A  change  of  the  hysteresis  cycle  occurs  only  at  very  slow  cycles 
— cycles  of  a  duration  from  several  minutes  to  years — and  even 
then  to  an  appreciable  extent  only  at  very  low  magnetic  densities. 
Thus  at  low  values  of  B — below  1000 — hysteresis  cycles  taken  by 
ballistic  galvanometer  are  liable  to  become  irregular  and  erratic, 
by  "  magnetic  creepage. "  For  most  practical  purposes,  however, 
this  may  be  neglected. 

37.  As  the  industrially  most  important  varying  magnetic  fields 
are  the  alternating  magnetic  fields,  the  hysteresis  loss  in  alternat- 
ing magnetic  fields,  that  is,  in  symmetrical  cycles,  is  of  most 
interest. 

In  general,  if  a  magnetic  flux  changes  from  the  condition  HI, 
B\:  point  PI  of  Fig.  29,  to  the  condition  H2,  B%:  point  P2,  and  we 
assume  this  magnetic  circuit  surrounded  by  an  electric  circuit  of 


58  ELECTRIC  CIRCUITS 

n  turns,  the  change  of  magnetic  flux  induces  in  the  electric  cir- 
cuit the  voltage,  in  absolute  units, 


it  is,  however, 

$  =  sB  (2) 

where  s  =  section  of  magnetic  circuit.     Hence 


If  i  =  current  in  the  electric  circuit,  the  m.m.f.  is 

F  =  ni  (4) 

and  the  magnetizing  force 

' 


where  I  =  length  of  the  magnetic  circuit. 
And  the  field  intensity 

H  =  47T/  (6) 

hence,  substituting  (5)  into  (6)  and  transposing, 

IH 
1  --  -^  (7) 

is  the  magnetizing  current  in  the  electric  circuit,  which  produces 
the  flux  density,  B. 

The  power  consumed  by  the  voltage  induced  in  the  electric 
circuit  thus  is 

slHdB 


or,  per  cm.3  of  the  magnetic  circuit,  that  is,  for  s  =  1  and  I 

H  dB 


and  the  energy  consumed  by  the  change  from  HI,  BI  to  H2,  B2, 
which  is  transferred  from  the  electric  into  the  magnetic  circuit, 
or  inversely, 


HdBergs  (10) 

. 
4r 


MAGNETISM 


59 


where  A\t  2  is  the  area  shown  shaded  in  Fig.  29. 

The  energy  consumed  during  a  cycle,  from  Ho,  BQ  to  —  Ho,  —  BQ 
and  back  to  H0,  B0,  thus  is 

-.      /»n 

(ii) 


w  =  7      I    HdB  ergs 


A 
=  T~  ergs 


(12) 


where 


r 


HdB  =  A  is  the  area  of  the  hysteresis  loop,  shown  shaded 


in  Fig.  30. 

As  the  magnetic  condition  at  the  end  of  the  cycle  is  the  same  as 


Hi 


FIG.  29. 


+  H+B 


Ha 


-H.-B 


FIG.  30. 


H,-B 


at  the  beginning,  all  this  energy,  w,  is  dissipated  as  heat,  that  is, 
is  the  hysteresis  energy  which  measures  the  molecular  magnetic 
friction. 

38.  If  in  Fig.  30  the  shaded  area  represents  the  hysteresis  loop 
between  +  H,  +  B,  and  —  H,  —  B,  giving  with  a  sinusoidal 
alternating  flux  the  voltage  and  current  waves,  Fig.  31,  the  maxi- 
mum area,  which  the  hysteresis  loop  could  theoretically  assume, 
is  given  by  the  rectangle  between  +  H,  +  B',  —  H,  +  B'}  —  H, 
—  B',  -\-  H,  —  B.  This  would  mean,  that  the  magnetic  flux  does 
not  appreciably  decrease  with  decreasing  field  intensity,  until 
the  field  has  reversed  to  full  value.  It  would  give  the  theoretical 
wave  shape  shown  as  Fig.  32.  As  seen,  this  is  the  extreme  ex- 
aggeration of  wave  shape,  Fig.  31. 


60 


ELECTRIC  CIRCUITS 


The  total  energy  of  this  rectangle,   or  maximum  available 
magnetic  energy,  is 

4HB       HB 

™°  =  --       "-  (12> 


D 

or,  if  /*  =  permeability,  thus  H  =  —  ,  it  is 


(13) 


FIG.  31. 


the  maximum  possible  hysteresis  loss. 

The  inefficiency  of  the  magnetic  cycle,  or  percentage  loss  of 
energy  in  the  magnetic  cycle,  thus  is 


FIG.  32. 


4B2 


HdB 


(14) 


39.  Experiment  shows  that  for  medium  flux  density,  that  is, 
thoses  values  of  B  which  are  of  the  most  importance  industrially, 


MAGNETISM 


61 


from  B  =  1000  to  B  =  12,000,  the  hysteresis  loss  can  with  suffi- 
cient accuracy  for  most  practical  purposes  be  approximated  by 
the  empirical  equation, 

w  =  -nB1-6  (15) 


/ 

/ 

0000 

SIL 

cor 

SI 

'EEL 

/ 

H> 

'STE 

RES 

IS 

9000 

7 

7 

8000 

7 

/ 

I 

/ 

7000 

I 

/ 

1  / 

'i' 

GOOD 

/ 

7 

// 

5000 

^ 

' 

^ 

' 

4000 

/ 

3000 

/ 

/ 

' 

2000 

/ 

f 

/ 

^ 

1000 

X 

B 

_^-< 

J^j 

!        I 

1 

>       ( 

i 

1 

$      { 

i      i 

0       1 

i     i 

2       1 

3      1 

4      1£ 

xlO* 

FIG.  33. 

where  77,  the  "coefficient  of  hysteresis,"  is  of  the  magnitude 
of  1  X  10-3  to  2  X  10-3  for  annealed  soft  sheet  steel,  if  B  is 
given  in  lines  of  force  per  cm.2,  and  w  is  ergs  per  cm.3  and  cycle. 
Very  often  w  is  given  in  joules,  or  watt-seconds  per  cycle  and 
per  kilogram  or  pound  of  iron,  and  B  in  lines  per  square  inch, 
or  w  is  given  in  watts  per  kilogram  or  per  pound  at  60  cycles. 


62 


ELECTRIC  CIRCUITS 


In  Fig.  33  is  shown,  with  B  as  abscissae,  the  hysteresis  loss,  w, 
of  a  sample  of  silicon  steel.  The  observed  values  are  marked 
by  circles.  In  dotted  lines  is  given  the  curve  calculated  by  the 
equation 

w  =  0.824  X  10-3  B1-6  (16) 

As  seen,  the  agreement  the  curve  of  1.6th  power  with  the  test 
values  is  good  up  to  B  =  10,000,  but  above  this  density,  the 
observed  values  rise  above  the  curve. 

40.  In  Fig.  34  is  plotted,  with  field  intensity,  H,  as  abscissas, 
the  magnetization  curve  of  ordinary  annealed  sheet  steel,  in 


FERRITE  AND  MAGNETITE 
MAGNETIZATION 


FIG.  34. 

half-scale,  as  curve  I,  and  the  magnetization  curve  of  magnetite, 
Fe3O4 — which  is  about  the  same  as  the  black  scale  of  iron — in 
double-scale,  as  curve  II.  As  III  then  is  plotted,  in  full-scale, 
a  curve  taking  0.8  of  I  and  0.2  of  II.  This  would  correspond  to 
the  average  magnetic  density  in  a  material  containing  80  per  cent, 
of  iron  and  20  per  cent,  (by  volume)  of  scale.  Curves  I'  and  III' 
show  the  initial  part  of  I  and  III,  with  ten  times  the  scale  of 
abscissae  and  the  same  scale  of  ordinates. 

Fig.  35  then  shows,  with  the  average  magnetic  flux  density,  B, 
taken  from  curve  III  of  Fig.  34,  as  abscissa,  the  part  of  the  mag- 


MAGNETISM 


63 


netic  flux  density  which  is  carried  by  the  magnetite,  as  curve  I. 
As  seen,  the  magnetite  carries  practically  no  flux  up  to  B  =  10, 
but  beyond  B  =  12,  the  flux  carried  by  the  magnetite  rapidly 


increases. 


As  curve  II  of  Fig.  35  is  shown  the  hysteresis  loss  in  this  inhomo- 
geneous  material  consisting  of  80  per  cent,  ferrite  (iron)  and  20 
per  cent,  magnetite  (scale)  calculated  from  curves  I  and  II  of  Fig. 


8000 

1 

6JU( 

8 

/ 

f 

1 

/ 

a 

7000 

1 

4 

III 

/ 

/ 

j 

g 

/ 

f 

COOO 

"/ 

/ 
/ 

// 

\ir 

5000 

/, 

PER 

RITI 

.    AND 

MAC 

NET 

ITE 

/ 

4000 

HYS 

TER 

ESIS 

/ 

/ 

lo'x 
f—& 

3000 

/ 

/ 

/ 

.5 

2CCO 

/ 

/ 

.4 

/ 

/ 

.3 

1000 

/ 

/ 

/ 

.2 

/ 

/ 

'/ 

xl 

)8   1 

- 

^ 

>      : 

J 

; 

; 

5 

r 

L_  1 

,      1 

- 

2—1 

^^ 

/ 

2      ] 

3      1 

4      15X10] 

FIG.  35. 

34  under  the  assumption  that  either  material  rigidly  follows  the 
1.6th  power  law  up  to  the  highest  densities,  by  the  equation, 
Iron: 

wi  =  1.2  JV'6  X  10-3. 
Scale: 

w2  =  23.5  Bn1'*  X  lO-3, 

As  curve  IF  is  shown  in  dotted  lines  the  1.6th  power  equation, 
w  =  1.38     B1'6  X  10~3. 


64  ELECTRIC  CIRCUITS 

As  seen,  while  either  constituent  follows  the  1.6th  power  law, 
the  combination  deviates  therefrom  at  high  densities,  and  gives 
an  increase  of  hysteresis  loss,  of  the  same  general  characteristic 
as  shown  with  the  silicon  steel  in  Fig.  33,  and  with  most  similar 
materials. 

As  curve  III  in  Fig.  35  is  then  shown  the  increase  of  the  hyste- 
resis coefficient  77,  at  high  densities,  over  the  value  1.38  X  10~3, 
which  it  has  at  medium  densities. 

Thus,  the  deviation  of  the  hysteresis  loss  at  high  densities/ 
from  the  1.6th  power  law,  may  possibly  be  only  apparent,  and 
the  result  of  lack  of  homogeneity  of  the  material. 

41.  At  low  magnetic  densities,  the  law  of  the  1.6th  power  must 
cease  to  represent  the  hysteresis  loss  even  approximately. 

The  hysteresis  loss,  as  fraction  of  the  available  magnetic  energy, 
is,  by  equation  (14), 


Substituting  herein  the  parabolic  equation  of  the  hysteresis 
loss, 

w  =  rjBn  (17) 

where  n  =  1.6,  it  is 

B"-2  (18) 

BA 


With  decreasing  density  B,Bn~2  steadily  increases,  if  n  <  2,  and 
as  the  permeability  //  approaches  a  constant  value,  f  ,  steadily  in- 
creases in  this  case,  thus  would  become  unity  at  some  low  density, 
B,  and  below  this,  greater  than  unity.  This,  however,  is  not 
possible,  as  it  would  imply  more  energy  dissipated,  than  available, 
and  thus  would  contradict  the  law  of  conservation  of  energy. 
Thus,  for  low  magnetic  densities,  if  the  parabolic  law  of  hysteresis 
(17)  applies,  the  exponent  must  be:  n  ^  2. 

In  the  case  of  Fig.  33,  for  rj  =  0.824  X  10~3,  assuming  the  per- 
meability for  extremely  low  density  as 

/x  =  1500, 

f  becomes  unity,  by  equation  (18),  at 

B  =  30. 

If  n  >  2,  Bn  ~  2  steadily  decreases  with  decreasing  B}  and  the  per- 
centage hysteresis  loss  becomes  less,  that  is,  the  cycle  approaches 
reversibility  for  decreasing  density;  in  other  words,  the  hys- 
teresis loss  vanishes.  This  is  possible,  but  not  probable,  and  the 


MAGNETISM 


65 


probability  is  that  for  very  low  magnetic  densities,  the  hysteresis 
losses  approach  proportionality  with  the  square  of  the  magnetic 
density,  that  is,  the  percentage  loss  approaches  constancy. 
From  equation  (17)  follows 


1.2 


.1.0. 


SILICON    STEEL 


HYSTER 


2.0 


AA 


LOG    B 


A  ; 


3.0 


4.0 


-3.0 


_2.0. 


-1.0J 


2.0 


FIG.  36. 

log  w  =  log  ri  +  n  log  B  (19) 

That  is: 

"If  the  hysteresis  loss  follows  a  parabolic  law,  the  curve  plotted 
with  log  w  against  log  B  is  a  straight  line,  and  the  slope  of  this 
straight  line  is  the  exponent,  n." 


66  ELECTRIC  CIRCUITS 

Thus,  to  investigate  the  hysteresis  law,  log  w  is  plotted  against 
log  B.  This  is  done  for  the  silicon  steel,  Fig.  33,  over  the  range 
from  B  =  30  to  B  =  16,000,  in  Fig.  36,  as  curve  I. 

Curve  I  contains  two  straight  parts,  for  medium  densities, 
from  log  B  =  3;  B  =  1000,  to  log  B  =  4;  B  =  10,000,  with  slope 
1.6006,  and  for  low  densities,  up  to  log  B  =  2.6;  B  =  400,  with 
slope  2.11.  Thus  it  is 

For         1000  <  B  <  10,000: 

w  =  0.824  B1'6  X  10~3 
For         B  <  400: 

w  =  0.00257  B2'11  X  10-3 

However,  in  this  lower  range,  n  =  2  gives  a  curve: 
w  =  0.0457  B2  X  10-3 

which  still  fairly  well  satisfies  the  observed  values. 

As  the  logarithmic  curve  for  a  sample  of  ordinary,  annealed 
sheet  steel,  Fig.  37,  gives  for  the  lower  range  the  exponent, 

n  =  1.923, 

and  as  the  difficulties  of  exact  measurements  of  hysteresis  losses 
increase  with  decreasing  density,  it  is  quite  possible  that  in  both, 
Figs.  36  and  37  the  true  exponent  in  the  lower  range  of  mag- 
netic densities  is  the  theoretically  most  probable  one, 

n  =  2, 

that  is,  that  at  about  B  =  500,  in  iron  the  point  is  reached,  below 
which  the  hysteresis  loss  varies  with  the  square  of  the  magnetic 
density. 

42.  As  over  most  of  the  magnetic  range  the  hysteresis  loss  can 
be  expressed  by  the  parabolic  law  (17),  it  appears  desirable  to 
adapt  this  empirical  law  also  to  the  range  where  the  logarithmic 
curve,  Figs.  36  and  37,  is  curved,  and  the  parabolic  law  does  not 
apply,  above  B  =  10,000,  and  between  B  =  500  and  B  =  1000, 
or  thereabouts.  This  can  be  done  either  by  assuming  the  coeffi- 
cient 77  as  variable,  or  by  assuming  the  exponent  n  as  variable. 

(a)  Assuming  77  as  constant, 

t]  =  0.824  X  10~3  for  the  medium  range,  where  n  =  1.6 
77!  =  0.0457  X  10-3  for  the  low  range,  where  HI  =  2 

The  coefficients  n  and  HI  calculated  from  the  observed  values 


MAGNETISM 


67 


of  w,  then,  are  shown  in  Fig.  36  by  the  three-cornered  stars  in 
the  upper  part  of  the  figure. 
(6)  Assuming  n  as  constant, 

n  =  1.6  for  the  medium  range,  where  77  =  0.0824  X  10~3 
n\  =  2  for  the  low  range,  where  r/i  =  0.0457  X  10~3 


10 


10  X-  -1^ 


1.9- 


1.7- 


1.4- 


1.1- 


ORDINARY  SHEET  STEEL,  ANNEALED 
HYSTERESIS 


2.0 


LOG   B 
80 


4.0 


-3.0 


-2.0 


-1.0 


-ua 


FIG.  37. 

The  variation  of  77  and  rji,  from  the  values  in  the  constant  range, 
then,  are  best  shown  in  per  cent.,  that  is,  the  loss  w  calculated  from 
the  parabolic  equation  and  a  correction  factor  applied  for  values 
of  B  outside  of  the  range. 


68  ELECTRIC  CIRCUITS 

Fig.  37  shows  the  values  of  rj  and  TJI,  as  calculated  from  the  para- 
bolic equations  with  n  =  1.6  and  HI  =  2,  and  Fig.  36  shows  the 
percentual  variation  of  17  and  771. 

The  latter  method,  (b),  is  preferable,  as  it  uses  only  one  expo- 
nent, 1.6,  in  the  industrial  range,  and  uses  merely  a  correction 
factor.  Furthermore,  in  the  method  (a),  the  variation  of  the 
exponent  is  very  small,  rising  only  to  1.64,  or  by  2.5  per  cent.,  while 
in  method  (b)  the  correction  factor  is  1.46,  or  46  per  cent.,  thus  a 
much  greater  accuracy  possible. 

43.  If  the  parabolic  law  applies, 

w  =  f]Bn  (17) 

the  slope  of  the  logarithmic  curve  is  the  exponent  n. 

If,  however,  the  parabolic  law  does  not  rigidly  apply,  the  slope 
of  the  logarithmic  curve  is  not  the  exponent,  and  in  the  range, 
where  the  logarithmic  curve  is  not  straight,  the  exponent  thus 
can  not  even  be  approximately  derived  from  the  slope. 

From  (17)  follows 

log  w  =  log  t\  +  n  log  B,  (19) 

differentiating  (19),  gives,  in  the  general  case,  where  the  parabolic 
law  does  not  strictly  apply, 

d  log  w  =  d  log  77  +  nd  log  B  +  log  Bdn, 
hence,  the  slope  of  the  logarithmic  curve  is 

d  log  w  L       n      dn  d  log  17  \  /0  , 

dW  -  n  +  (log  B  JW  +  Jiie) 

If  n  =  constant,  and  t]  =  constant,  the  second  term  on  the 
right-hand  side  disappears,  and  it  is 

d  log  w 


that  is,  the  slope  of  the  logarithmic  curve  is  the  exponent. 

If,  however,  77  and  n  are  not  constant,  the  second  term  on  the 
right-hand  side  of  equation  (20)  does  not  in  general  disappear, 
and  the  slope  thus  does  not  give  the  exponent. 

Assuming  in  this  latter  case  the  slope  as  the  exponent,  it  must 
be 

1mr  p      dn  dlogrj     _ 

l°gBdA^B^      d\^B  ~    °' 
Or, 

=-log*  (22) 


MAGNETISM 


69 


In  this  case,  n  and  much  more  still  TJ  show  a  very  great  varia- 
tion, and  the  variation  of  77  is  so  enormous  as  to  make  this  repre- 
sentation valueless. 

As  illustration  is  shown,  in  Fig.  36,  the  slope  of  the  curve  as 
ri2.  As  seen,  nz  varies  very  much  more  than  n  or  n\. 

To  show  the  three  different  representations,  in  the  following 
table  the  values  of  n  and  t\  are  shown,  for  a  different  sample  of 
iron. 

TABLE 


B  103 

(a)  »j  =  const. 
1  2^4 

(b)  n  =  const.  = 
1  fi 

(c)  n,  = 
a  log  w 

'» 

dlogB 

below   10.00 

n=    .6 

77  =  1.254X100~3 

tt2  =  1.6 

772  =  1.  254X10-' 

10.00 

=    .601 

=  1.268 

=  1.79 

230.00 

11.23 

=    .604 

=  1.302 

=  2.23 

3.68 

12.63 

=    .617 

=  1  .  468 

=  2.66 

0.0488 

13.30 

=    .624 

=  1.570 

=  2.83 

0.0133 

14.00 

=  1.630 

=  1.668 

=  2.98 

0.0032 

14.65 

=  1  .  634 

=  1  .  738 

=  3.15 

0.00069 

1  738 

As  seen,  to  represent  an  increase  of  hysteresis  loss  by  ^-^^  = 

1  .^o4 

1.39,  or  39  per  cent.,  under  (c),  n2  is  nearly  doubled,  and  772  re- 
duced to  .,  ortn  fw*  of  its  initial  value. 

l,oUU,UUU 

44.  The  equation  of  the  hysteresis  loss  at  medium  densities, 
W  =  TjBn;  n  =  1.6 

is  entirely  empirical,  and  no  rational  reason  has  yet  been  found 
why  this  approximation  should  apply.  Calculating  the  coeffi- 
cient n  from  test  values  of  B  and  W,  shows  usually  values  close  to 
1.6,  but  not  infrequently  values  of  n  are  found,  as  low  as  1.55,  and 
even  values  below  1.5,  and  values  up  to  1.7  and  even  above  1.9 
In  general,  however,  the  more  accurate  tests  give  values  of  n 
which  do  not  differ  very  much  from  1.6,  so  that  the  losses  can 
still  be  represented  by  the  curve  with  the  exponent  n  =  1.6, 
without  serious  error.  This  is  desirable,  as  it  permits  comparing 
different  materials  by  comparing  the  coefficients  77.  This  would 
not  be  the  case,  if  different  values  of  n  were  used,  as  even  a  small 
change  of  n  makes  a  very  large  change  of  rj :  a  change  of  n  by  1  per 
cent.,  at  B  =  10,000,  changes  77  by  about  16  per  cent. 


70 


ELECTRIC  CIRCUITS 


Thus  in  Fig.  37  is  represented  as  I  the  logarithmic  curve  of  a 
sample  of  ordinary  annealed  sheet  steel,  which  at  medium  den- 
sity gives  the  exponent  n  =  1.556,  at  low  densities  the  exponent 
HI  =  1.923.  Assuming,  however,  n  =  1.6  and  HI  =  2.0,  gives 
the  average  values  77  =  1.21  X  10~3  and  771  =  0.10  X  10~3,  and  the 


10 

tooL 

ORDINARY    SHEET    STEEL, 
ANNEALED.      HYSTERESIS 

/ 

9000 

/ 

/ 

8000 

/ 

/ 

7000 

/ 

/ 

GO 

y- 

/ 

/ 
/ 

j 

/ 

/ 

/ 

5000 

/ 

/ 

/ 
/ 

/ 

/ 
/ 

4000 

/ 

/ 

;. 

£/ 

3000 

/' 

^ 

/ 

2000 

/ 

/ 

^ 

S 

j 

1000 

^ 

/ 

-=e: 

s£ 

r** 
'- 

J        * 

\ 

> 

' 

! 

B 

\       ' 

\     i 

D      1 

1       1 

2      1 

3       1 

4       1 

5x10 

FIG.  38. 

individual  calculated  values  of  rj  and  771  are  then  shown  on  Fig. 
37  by  crosses  and  three-pointed  stars,  respectively. 

Fig.  38  then  shows  the  curve  of  observed  loss,  in  drawn  line, 
and  the  1.6th  power  curve  calculated  in  dotted  line,  and  Fig.  39 
the  lower  range  of  the  calculated  curve,  with  the  observations 
marked  by  circles.  Fig.  40  shows,  for  the  low  range,  the  curve 


MAGNETISM 


71 


of  rjiB2,  in  two  different  scales,  with  the  observed  values  marked 
by  cycles.  As  seen,  although  in  this  case  the  deviation  of  n  from 
1.6  respectively  2  is  considerable,  the  curves  drawn  with  n  = 
1.6  and  Wi  =  2  still  represent  the  observed  values  fairly  well  in 


7 

ORDINARY    SHEET    STEEL, 
ANNEALED.      HYSTERESIS 
MEDIUM    DENSITIES 

/ 

100Q 

/ 

_900 

/ 

I 

800 

1 

1 

_700 

/ 

/ 

600 

/ 

/ 

500 

/ 

/ 

400 

/ 

i 

/ 

300 

/ 

\ 

/ 

200 

/ 

/ 

-100 

^ 

f 

.> 

1* 

i 

1 

B 

i 

| 

xlO8 

FIG.  39. 


the  range  of  B  from  500  to  10,000,  and  below  500,  respectively,  so 
that  the  1.6th  power  equation  for  the  medium,  and  the  quadratic 
equation  for  the  low  values  of  B  can  be  assumed  as  sufficiently 
accurate  for  most  purposes,  except  in  the  range  of  high  densities 


72 


ELECTRIC  CIRCUITS 


in  those  materials,  where  the  increase  of  hysteresis  loss  occurs 
there. 

While  the  measurement  of  the  hysteresis  loss  appears  a  very 
simple  matter,  and  can  be  carried  out  fairly  accurately  over  a 


/ 

ORDINARY    SHEET    STEEL, 
ANNEALED.      HYSTERESIS 
LOW    DENSITIES 

/ 

/ 

~T 

/ 

/ 

32 

/' 

80 

1 

28 

/ 

26 

/ 

I 

_24 

/ 

22 

/ 

20 

/ 

18 

1.6 

/ 

/ 

16 

1.4 

/ 

/ 

> 

14 

1.2 

/ 

/ 

12 

1.0 

< 

/ 

/ 

10 

..8 

/ 

/ 

/ 
| 

8 

.6 

/ 

6 

/ 

6 

.4 

/ 

/ 

4 

.?,  j 

/ 

.X 

^ 

B 

2 

I 

•*3 

S* 

J 

! 

g 

: 

• 

» 

. 

jxlOS 

FIG.  40. 


narrow  range  of  densities,  it  is  one  of  the  most  difficult  matters  to 
measure  the  hysteresis  loss  over  a  wide  range  of  densities  with 
such  accuracy  as  to  definitely  determine  the  exact  value  of  the 
exponent  n,  due  to  varying  constant  errors,  which  are  beyond  con- 


MAGNETISM  73 

trol.  While  true  errors  of  observations  can  be  eliminated  by 
multiplying  data,  with  a  constant  error  this  is  not  the  case,  and  if 
the  constant  error  changes  with  the  magnetic  density,  it  results 
in  an  apparent  change  of  n.  Such  constant  errors,  which  increase 
or  decrease,  or  even  reverse  with  changing  B,  are  in  the  Ballistic 
galvanometer  method  the  magnetic  creepage  at  lower  B,  and  at 
higher  B  the  sharp-pointed  shape  of  the  hysteresis  loop,  which 
makes  the  area  between  rising  and  decreasing  characteristic 
difficult  to  determine.  In  the  wattmeter  method  by  alternating 
current,  varying  constant  errors  are  the  losses  in  the  instruments, 
the  eddy-current  losses  which  change  with  the  changing  flux  dis- 
tribution by  magnetic  screening  in  the  iron,  with  the  temperature, 
etc.,  by  wave-shape  distortion,  the  unequality  of  the  inner  and 
outer  length  of  the  magnetic  circuit,  etc. 

45.  Symmetrical  magnetic  cycles,  that  is,  cycles  performed  be- 
tween equal  but  opposite  magnetic  flux  densities,  -\-B  and  —  Bt 
are  industrially  the  most  important,  as  they  occur  in  practically 
all  alternating-current  apparatus.  Unsymmetrical  cycles,  that 
is,  cycles  between  two  different  values  of  magnetic  flux  density, 
BI  and  J52,  which  may  be  of  different,  or  may  be  of  the  same 
sign,  are  of  lesser  industrial  importance,  and  therefore  have  been 
little  investigated  until  recently. 

However,  unsymmetrical  cycles  are  met  in  many  cases  in  al- 
ternating- and  direct-current  apparatus,  and  therefore  are  of 
importance  also. 

In  most  inductor  alternators  the  magnetic  flux  in  the  armature 
does  not  reverse,  but  pulsates  between  a  high  and  a  low  value  in 
the  same  direction,  and  the  hysteresis  loss  thus  is  that  of  an 
unsymmetrical  non-reversing  cycle. 

Unsymmetrical  cycles  occur  in  transformers  and  reactors  by  the 
superposition  of  a  direct  current  upon  the  alternating  current,  as 
discussed  in  the  chapter  " Shaping  of  Waves,"  or  by  the  equiva- 
lent thereof,  such  as  the  suppression  of  one-half  wave  of  the  alter- 
nating current.  Thus,  in  the  transformers  and  reactors  of  many 
types  of  rectifiers,  as  the  mercury-arc  rectifier,  the  magnetic  cycle 
is  unsymmetrical. 

Unsymmetrical  cycles  occur  in  certain  connections  of  trans- 
formers (three-phase  star-connection)  feeding  three-wire  syn- 
chronous converters,  if  the  direct-current  neutral  of  the  converter 
is  connected  to  the  transformer  neutral. 

They  may  occur  and  cause  serious  heating,  if  several  trans- 


74  ELECTRIC  CIRCUITS 

formers  with  grounded  neutrals  feed  the  same  three-wire  distri- 
bution circuit,  by  stray  railway  return  current  entering  the  three- 
wire  a  ternating  distribution  circuit  over  one  neutral  and  leaving 
it  over  another  one. 

Two  smaller  unsymmetrical  cycles  often  are  superimposed  on 
an  alternating  cycle,  and  then  increase  the  hysteresis  loss.  Such 
occurs  in  transformers  or  reactors  by  wave  shapes  of  impressed 
voltage  having  more  than  two  zero  values  per  cycle,  such  as  that 
shown  in  Fig.  51  of  the  chapter  on  "Shaping  of  Waves." 

They  also  occur  sometimes  in  the  armatures  of  direct-current 
motors  at  high  armature  reaction  and  low  field  excitation,  due  to 
the  flux  distortion,  and  under  certain  conditions  in  the  armatures 
of  regulating  pole  converters. 

A  large  number  of  small  unsymmetrical  cycles  are  sometimes 
superimposed  upon  the  alternating  cycle  by  high-frequency  pul- 
sation of  the  alternating  flux  due  to  the  rotor  and  stator  teeth, 
and  then  may  produce  high  losses.  Such,  for  instance,  is  the 
case  in  induction  machines,  if  the  stator  and  rotor  teeth  are  not 
proportioned  so  as  to  maintain  uniform  reluctance,  or  in  alterna- 
tors or  direct-current  machines,  in  which  the  pole  faces  are  slotted 
to  receive  damping  windings,  or  compensating  windings,  etc., 
if  the  proportion  of  armature  and  pole-piece  slots  is  not  carefully 
designed. 

46.  The  hysteresis  loss  in  an  unsymmetrical  cycle,  between 
limits  BI  and  B2,  that  is,  with  the  amplitude  of  magnetic  variation 

7?          _     T) 

B  =  --^-~ — -,  follows  the  same  approximate  law  of  the  1.6th 
power, 

as  long  as  the  average  value  of  the  magnetic  flux  variation, 


2 

is  constant. 

With  changing  B0,  however,  the  coefficient  r)0  changes,  and  in- 
creases with  increasing  average  flux  density,  BQ. 

John  D.  Ball  has  shown,  that  the  hysteresis  coefficient  of  the 
unsymmetrical  cycle  increases  with  increasing  average  density, 
BQ,  and  approximately  proportional  to  a  power  of  BQ.  That  is, 

^  =  ^  +  fa  £01.9. 


MAGNETISM  75 

Thus,  in  an  unsymmetrical  cycle  between  limits  BI  and  B2  of 
magnetic  flux  density,  it  is 


w  = 


where  rj  is  the  coefficient  of  hysteresis  of  the  alternating-current 
cycle,  and  for  B%  =  —  Bi,  equation  (23)  changes  to  that  of  the 
symmetrical  cycle. 
Or,  if  we  substitute, 

Bo  =  ^±^?  (24) 

=  average  value  of  flux  density,  that 
is,  average  of  maximum  and  mini- 
mum. 


(25) 


=  amplitude  of  unsymmetrical  cycle, 
it  is 

w  =  (77+  jSBo1-9)*1'6  (26) 

or, 

w  =  r/oB1-6  (27) 

where 

7,0  =  77  +  W'9  (28) 

or,  more  general, 

w  =  rjQBn  (29) 

T/o    =    7,   +   jS^o"  (30) 

For  a  good  sample  of  ordinary  annealed  sheet  steel,  it  was 
found, 

rj  =  1.06  X  10-3  (31) 

j8  =  0.344  X  10-10 
For  a  sample  of  annealed  medium  silicon  steel, 

77  =  1.05  X  10-3 

(32) 
0  =  0.32  X  10-10 

Fig.  41  shows,  with  B0  as  abscissae,  the  values  of  T?O,  by  equa- 
tions (30)  and  (32). 

As  seen,  in  a  moderately  unsymmetrical  cycle,  such  as  between 
BI  =  +12,000  and  Bz  =  —4000,  the  increase  of  the  hysteresis 


76 


ELECTRIC  CIRCUITS 


loss  over  that  in  a  symmetrical  cycle  of  the  same  amplitude,  is 
moderate,  but  the  increase  of  hysteresis  loss  becomes  very  large 


°"> 

A 

/ 

'  3,8 

UNSYMMETRICAL  CYCLE 
770-1.05xlO"3-t-.32B1*9xlO"8 

/ 

36 

/ 

/ 

34 

/ 

3.2 

/ 

3.0 

/ 

/ 

fl,R 

/ 

flfi 

/ 

2.4 

/ 

2.2 

/ 

f 

«n 

/ 

/ 

1.8 

/ 

/ 

1.6 

/ 

1  4 

^ 

/ 

1  fl 

— 

—  -•* 

*^ 

1.0 

.8 

.6 

.4 

?, 

• 

1 

\      \ 

\             L 

;     i 

,     • 

\ 

\     \ 

)     i 

5     \ 

i     i 

2      1 

3      1 

4       1 

5X108 

FIG.  41. 


in  highly  unsymmetrical  cycles,  such  as  between  B\  =   16,000 
and  B2  =  12,000. 


CHAPTER  V 

MAGNETISM 

Magnetic  Constants 

47.  With  the  exception  of  a  few  ferromagnetic  substances,  the 
magnetic  permeability  of  all  materials,  conductors  and  dielectrics, 
gases,  liquids  and  solids,  is  practically  unity  for  all  industrial 
purposes.  Even  liquid  oxygen,  which  has  the  highest  permea- 
bility, differs  only  by  a  fraction  of  a  per  cent,  from  non-magnetic 
materials. 

Thus  the  permeability  of  neodymium,  which  is  one  of  the  most 
paramagnetic  metals,  is  n  =  1.003;  the  permeability  of  bismuth, 
which  is  very  strongly  diamagnetic,  is  /*  =  1  —  0.00017  =  0.99983. 

The  magnetic  elements  are  iron,  cobalt,  nickel,  manganese 
and  chromium.  It  is  interesting  to  note  that  they  are  in  atomic 
weight  adjoining  each  other,  in  the  latter  part  of  the  first  half  of 
the  first  large  series  of  the  periodic  system: 

Ti     V     Cr    Mn   Fe    Co    Ni    Cu    Zn 
Atomic  weight 48     51     52     55     56     58     59     61     65 

The  most  characteristic,  because  relatively  most  constant,  is 
the  metallic  magnetic  saturation,  S,  or  its  reciprocal,  the  satura- 
tion coefficient,  a,  in  the  reluctivity  equation.  The  saturation 
density  seems  to  be  little  if  any  affected  by  the  physical  condition 
of  the  material.  By  the  chemical  composition,  such  as  by  the 
presence  of  impurities,  it  is  affected  only  in  so  far  as  it  is  reduced 
approximately  in  proportion  to  the  volume  occupied  by  the  non- 
magnetic materials,  except  in  those  cases  where  new  compounds 
result. 

It  seems,  that  the  saturation  value  is  an  absolute  limit  of  the 
element,  and  in  any  mixture,  alloy  or  compound,  the  saturation 
value  reduced  to  the  volume  of  the  magnetic  metal  contained 
therein,  can  not  exceed  that  of  the  magnetic  metal,  but  may  be 
lower,  if  the  magnetic  metal  partly  or  wholly  enters  a  compound 
of  lower  intrinsic  saturation  value.  Thus,  if  S  =  21  X  103  is 
the  saturation  value  of  iron,  an  alloy  or  compound  containing 

77 


78  ELECTRIC  CIRCUITS 

72  per  cent,  by  volume  of  iron  can  have  a  maximum  saturation 
value  of  S  =  0.72  X  21  X  103  =  15.1  X  103  only,  or  a  still  lower 
saturation  value. 

The  only  known  exception  herefrom  seems  to  be  an  iron-cobalt 
alloy,  which  is  alleged  to  have  a  saturation  value  about  10  per 
cent,  higher  than  that  of  iron,  though  cobalt  is  lower  than  iron. 

The  coefficient  of  magnetic  hardness,  a,  however,  and  the  co- 
efficient of  hysteresis,  17,  vary  with  the  chemical,  and  more  still 
with  the  physical  characteristic  of  the  magnetic  material,  over  an 
enormous  range. 

Thus,  a  special  high-silicon  steel,  and  the  chilled  glass  hard 
tool  steel  in  the  following  tables,  have  about  the  same  percentage 
of  non-magnetic  constituents,  4  per  cent.,  and  about  the  same 
saturation  value,  S  =  19.2  X  103,  but  the  coefficient  of  hardness 
of  chilled  tool  steel,  a  =  8  X  10~3,  is  200  times  that  of  the  special 
silicon  steel,  a  =  0.04  X  10~3,  and  the  coefficient  of  hysteresis  of 
the  chilled  tool  steel,  77  =  75  X  10~3,  is  125  times  that  of  the  sili- 
con steel,  i)  =  0.6  X  10~3.  Hardness  and  hysteresis  loss  seem 
to  depend  in  general  on  the  physical  characteristics  of  the  material, 
and  on  the  chemical  constitution  only  as  far  as  it  affects  the  phys- 
ical characteristics. 

Chemical  compounds  of  magnetic  metals  are  in  general  not 
ferromagnetic,  except  a  few  compounds  as  magnetite,  which  are 
ferromagnetic. 

With  increasing  temperature,  the  magnetic  hardness  a,  decreases, 
that  is,  the  material  becomes  magnetically  softer,  and  the  satura- 
tion density,  S,  also  slowly  decreases,  until  a  certain  critical 
temperature  is  reached  (about  760°C.  with  iron),  at  which  the 
material  suddenly  ceases  to  be  magnetizable  or  ferromagnetic, 
but  usually  remains  slightly  paramagnetic. 

As  the  result  of  the  increasing  magnetic  softness  and  decreasing 
saturation  density,  with  increasing  temperature  the  density, 
B,  at  low  field  intensities,  H,  increases,  at  high  field  intensities 
decreases.  Such  5-temperature  curves  at  constant  H,  however, 
have  little  significance,  as  they  combine  the  effect  of  two  changes, 
the  increase  of  softness,  which  predominates  at  low  H,  and  the 
decrease  of  saturation,  which  predominates  at  high  H. 

Heat  treatment,  such  as  annealing,  cooling,  etc.,  very  greatly 
changes  the  magnetic  constants,  especially  a  and  t\ — more  or 
less  in  correspondence  with  the  change  of  the  physical  constants 
brought  about  by  the  heat  treatment. 


MAGNETISM  79 

Very  extended  exposure  to  moderate  temperature — 100  to 
200° C. — increases  hardness  and  hysteresis  loss  with  some  mate- 
rials, by  what  is  called  ageing,  while  other  materials  are  almost 
free  of  ageing. 

48.  The  most  important,  and  therefore  most  completely  in- 
vestigated magnetic  metal  is  iron. 

Its  saturation  value  is  probably  between  S  =  21.0  X  103  and 
S  =  21.5  X  103,  the  saturation  coefficient  thus  a-  =  0.047. 

As  all  industrially  used  iron  contains  some  impurities, 
carbon,  silicon,  manganese,  phosphorus,  sulphur,  etc.,  usually 
saturation  values  between  20  X  103  and  21  X  103  are  found  on 
sheet  steel  or  cast  steel,  etc.,  lower  values,  19  to  19.5  X  103, 
in  silicon  steels  containing  several  per  cent,  of  Si,  and  still  much 
lower  values,  12  to  15  X  103,  in  very  impure  materials,  such  as 
cast  iron. 

Two  types  of  iron  alloys  seem  to  exist : 

1.  Those  in  which  the  alloying  material  does  not  directly  affect 
the  magnetic  qualities,  but  only  indirectly,  by  reducing  the  vol- 
ume of  the  iron  and  thereby  the  saturation  value,  and  by  chang- 
ing the  physical  characteristics  and  thereby  the  hardness  and 
hysteresis  loss. 

Such  apparently  are  the  alloys  with  carbon,  silicon,  titanium, 
chromium,  molybdenum  and  tungsten,  etc.,  as  oast  iron,  silicon 
steel,  magnet  steel,  etc. 

2.  Those  in  which  the  alloying  material  changes  the  magnetic, 
characteristics. 

Such  apparently  are  the  alloys  with  nickel,  manganese,  mercury, 
copper,  cobalt,  etc. 

In  this  class  also  belong  the  chemical  compounds  of  the  mag- 
netic materials. 

Thus,  a  manganese  content  of  10  to  15  per  cent,  makes  the  iron 
practically  non-magnetic,  lowers  the  permeability  to  /x  =  1.4. 
However,  even  here  it  is  not  certain  whether  this  is  not  an 
extreme  case  of  magnetic  hardness,  and  at  extremely  high 
magnetic  fields  the  normal  saturation  value  of  the  iron  would  be 
approached. 

Some  nickel  steels  (25  per  cent.  Ni)  may  be  either  magnetic,  or 
non-magnetic.  However,  pure  iron,  when  heated  to  high  incan- 
descence, becomes  non-magnetic  at  a  certain  definite  temperature, 
and  when  cooling  down,  becomes  magnetizable  again  at  another 
definite,  though  lower  temperature,  and  between  these  two  tern- 


80  ELECTRIC  CIRCUITS 

peratures,  iron  may  be  magnetic  or  unmagnetic,  depending 
whether  it  has  reached  this  temperature  from  lower,  or  from  higher 
temperatures.  Apparently,  for  these  nickel  steels,  the  critical 
temperature  range,  within  which  they  can  be  magnetic  or  un- 
magnetic, is  within  the  range  of  atmospheric  temperature,  and 
thus,  after  heating,  they  become  non-rnagnetic,  after  cooling  to 
sufficiently  low  temperature,  they  become  magnetizable  again. 
Thus,  a  steel  containing  17  per  cent,  nickel,  4.5  per  cent,  chro- 
mium, 3  per  cent,  manganese,  has  permeability  1.004,  that  is,  is 
almost  completely  unmagnetic. 

Heterogeneous  mixtures,  such  as  powdered  iron  incorporated 
in  resin,  or  iron  filings  in  air,  seem  to  give  saturation  densities 
not  far  different  from  those  corresponding  to  their  volume  per- 
centage of  iron,  but  give  an  enormous  increase  of  hardness,  a,  and 
hysteresis,  77,  as  is  to  be  expected. 

Most  chemical  compounds  of  iron  are  non-magnetic.  Fer- 
romagnetic is  only  magnetite,  which  is  the  intermediate  oxide 
and  may  be  considered  as  ferrous  ferrite.  There  also  is  an 
alleged  magnetic  sulphide  of  iron,  though  I  have  never  seen 
it,  magnetkies,  FeySg  or  FesS9. 

As  magnetite,  Fe304,  contains  72  per  cent,  of  Fe,  by  weight, 
and  has  the  specific  weight  5.1,  its  volume  per  cent,  of  iron  would 
be  48  per  cent.,  and  the  saturation  density  S  =  10  X  103. 

Observations  on  the  magnetic  constants  of  magnetite  give  a 
saturation  density  of  4.7  X  103  to  5.91  X  103,  so  that  magnet- 
ite would  fall  in  the  second  class  of  iron  compounds,  those  in 
which  the  saturation  density  is  affected,  and  lowered,  by  the 
composition. 

Not  only  magnetite,  which  may  be  considered  as  ferrous  ferrite, 
but  numerous  other  ferrites,  that  is,  salts  of  the  acid  Fe2O4H2, 
are  to  some  extent  ferromagnetic,  such  as  copper  and  cobalt  fer- 
rite, calcium  ferrite,  etc. 

49.  Cobalt,  next  adjoining  to  iron  in  the  periodic  system  of  ele- 
ments, is  the  magnetic  metal  which  has  been  least  investigated. 
Its  saturation  value  probably  is  between  S  =  12  X  103  and  S  = 
14  X  103,  and  its  magnetic  characteristic  looks  very  similar  to  that 
of  cast  iron.  Partly  this  is  due  to  the  similar  saturation  value, 
partly  probably  due  to  the  feature  that  most  of  the  available 
data  were  taken  on  cast  cobalt. 

It  is  interesting  to  note  that  Cobalt  retains  its  magnetizability 


MAGNETISM  81 

up  to  much  higher  temperatures  than  iron  or  any  other  material, 
so  that  above  800  degrees  C.,  Cobalt  is  the  only  magnetic  material. 

More  information  is  available  on  nickel,  the  metal  next  ad- 
joining to  cobalt  in  the  periodic  system  of  elements.  Its  satura- 
tion density  is  the  lowest  of  the  magnetic  metals,  probably  be- 
tween S  =  6  X  103  and  8  =  7  X  103. 

Some  data  on  nickel  and  nickel  alloys  are  given  in  the  following 
table.  In  general,  nickel  seems  to  show  characteristics  very  simi- 
lar to  those  of  iron,  except  that  all  the  magnetic  densities  are  re- 
duced in  proportion  to  the  lower  saturation  density;  but  the  effect 
of  the  physical  characteristics  on  the  magnetic  constants  appears 
to  be  the  same.  Interesting  is,  that  nickel  seems  to  be  least  sen- 
sitive to  impurities  in  their  effect  on  the  reluctivity  curve. 

Nickel  ceases  to  be  magnetizable  already  below  red  heat. 

The  next  metal  beyond  nickel,  in  the  periodic  system  of  ele- 
ments, is  copper,  and  this  is  non-magnetic,  as  far  as  known. 

On  the  other  side  of  iron,  in  the  periodic  system,  is  manganese. 

This  is  very  interesting  in  so  far  as  it  has  never  been  observed 
in  a  strongly  magnetic  state,  but  many  of  the  alloys  of  manganese 
are  more  or  less  strongly  magnetic,  and  estimating  from  the  satu- 
ration values  of  manganese  alloys,  the  saturation  value  of  man- 
ganese as  pure  metal  should  be  about  S  =  30  X  103.  This 
would  make  it  the  most  magnetic  metal. 

In  favor  of  manganese  as  magnetic  metal  also  is  the  unusual 
behavior  of  its  alloys  with  iron :  the  alloys  of  nickel,  and  of  cobalt 
with  iron  also  show  unusual  characteristics,  and  this  seems  to  be  a 
characteristic  of  alloys  between  magnetic  metals. 

The  best  known  magnetic  manganese  alloys  are  the  Heusler 
alloys,  of  manganese  with  copper  and  aluminum,  and  the  char- 
acteristics of  three  such  alloys  are  given  in  the  following  table. 
The  most  magnetic  shows  about  the  same  saturation  value  as 
magnetite,  but  higher  saturation  values,  equal  to  those  of  nickel, 
have  been  observed. 

A  curious  feature  of  some  Heusler  alloys  is,  that  when  slowly 
cooled  from  high  temperatures,  they  are  very  little  magnetic, 
and  have  low  saturation  values.  The  quicker  they  are  cooled, 
the  higher  their  permeability  and  their  saturation  value,  and  the 
best  values  have  been  reached  by  dropping  the  molten  alloy  into 
water,  so  suddenly  chilling  it. 

In  general,  the  Heusler  alloys  are  especially  sensitive  to  heat 
treatment,  and  some  of  them  show  the  ageing  in  a  most  pro- 


82  ELECTRIC  CIRCUITS 

nounced  degree,  so  that  maintaining  the  alloy  for  a  considerable 
time  at  moderate  temperature,  increases  hardness  and  hysteresis 
loss  more  than  tenfold. 

Magnetic  alloys  of  manganese  also  are  known  with  antimony, 
arsenic,  phosphorus,  bismuth,  boron,  with  zinc  and  with  tin,  etc. 
Usually,  the  best  results  are  given  by  alloys  containing  20  to  30 
per  cent,  of  manganese.  Little  is  known  of  these  magnetic  al- 
loys, except  that  they  may  be  in  a  magnetic  state,  or  in  an 
unmagnetic  stage.  They  are  most  conveniently  produced  by 
dissolving  manganese  metal  in  the  superheated  alloying  metal, 
or  in  this  metal  with  the  addition  of  some  powerful  reducing 
metal,  as  sodium  or  aluminum,  but  the  alloy  is  only  sometimes 
magnetic,  sometimes  practically  unmagnetic,  and  the  conditions 
of  the  formation  of  the  magnetic  state  are  unknown. 

Apparently,  there  also  exists  an  intermediary  oxide  of  mangan- 
ese, or  a  compound  oxide  of  manganese  with  that  of  the  other 
metal,  which  is  strongly  magnetic.  The  black  slag,  appearing  in 
the  fusion  of  manganese  with  other  metals  such  as  antimony, 
zinc,  tin,  without  flux,  often  is  strongly  magnetic,  more  so  than 
the  alloy  itself. 

A  mixture  of  about  25  per  cent,  powdered  manganese  metal, 
and  75  per  cent,  powdered  antimony  metal,  heated  together  to  a 
moderate  temperature — in  a  test-tube — gives  a  strongly  mag- 
netic black  powder,  which  can  be  used  like  iron  filings,  to  show 
the  lines  of  forces  of  the  magnetic  field,  but  has  not  further  been 
investigated. 

A  considerable  number  of  such  magnetic  manganese  alloys  have 
been  investigated  by  Heusler  and  others,  and  their  constants  are 
given  in  the  following  table. 

It  is  supposed  that  these  magnetic  manganese  alloys  are  chem- 
ical compounds,  similar  as  magnetite  or  magnetkies.  Thus  the 
copper-aluminum-manganese  alloy  of  Heusler  is  a  compound  of 
1  atom  of  aluminum  with  3  atoms  of  copper  or  manganese:  Al- 
(Mn  or  Cu)3,  usually  AlMnCu2.  Other  magnetic  manganese 
compounds  then  are: 

With  antimony MnSb  and  Mn2Sb 

With  bismuth MnBi 

With  arsenic MnAs 

With  boron MnB 

With  phosphorus MnP 

With  tin. ; Mn4Sn  and  Mn2Sn 


MAGNETISM  83 

Next  adjacent  to  manganese  in  the  periodic  system  of  elements 
is  chromium.  Neither  the  metal,  nor  any  of  its  alloys  (except 
those  with  magnetic  metals)  have  ever  been  observed  in  the  mag- 
netic state.  There  is,  however,  an  intermediary  oxide  of  chro- 
mium, alleged  to  be  Cr5O9  (a  basic  chromic  chromate?)  which  is 
strongly  magnetic.  It  forms,  in  black  scales,  in  a  narrow  range 
of  temperature,  by  passing  CrC^CU  with  hydrogen  through  a 
heated  tube. 

A  second  strongly  magnetic  chromium  oxide  is  Cr4O9  (a  basic 
chromic  bichromate?).  It  is  easily  produced  by  rapidly  heat- 
ing Cr03,  but  the  product  is  not  always  the  same.  Their 
magnetic  characteristics  have  never  been  investigated,  and  they 
are  the  only  indication  which  would  point  to  chromium  having 
potentially  magnetic  qualities. 

The  metal  next  to  chromium  in  the  periodic  system  of  elements, 
vanadium,  is  non-magnetic,  as  far  as  known. 

50.  On  attached  tables  are  given  the  magnetic  constants  of  the 
better  known  magnetic  materials,  metals,  alloys,  mixtures  and 
compounds: 

The  first  tables  give  the  saturation  density,  S,  and  the  demag- 
netization temperature,  that  is,  temperature  at  which  the  ma- 
terial ceases  to  be  ferromagnetic,  and  its  specific  gravity. 

It  is  interesting  to  note  that  with  some  magnetic  materials  the 
demagnetization  temperature  is  very  close  to,  or  within  the  range 
of,  atmospheric  temperature. 

The  second  table  gives  more  complete  data  of  those  materials, 
of  which  such  data  are  available.  It  gives: 

S  =  saturation  density,  or  value  of  B  —  H  for  infinitely  high  H\ 

a  =  coefficient  of  magnetic  hardness; 

<r  =  coefficient  of  magnetic  saturation. 

Where  the  reluctivity  line  shows  a  bend  at  some  critical  point, 
a  and  or  are  given  for  the  lower  range — which  is  the  one  indus- 
trially most  useful — together  with  the  range  of  field  intensity,  for 
which  this  value  applies,  and  are  given  also  for  the  highest  range 
observed,  together  with  the  value  of  field  intensity  //,  above 
which  the  latter  values  of  a  and  a  apply. 

77  =  coefficient  of  hysteresis,  in  the  1.6th  power  law. 

/3  =  coefficient  of  unsymmetrical  cycle,  for  the  two  cases 
where  this  is  known. 

Demagnetization  temperature,  that  is,  temperature  at  which 
ferromagnetism  ceases. 


84  ELECTRIC  CIRCUITS 

p  =  electrical  resistivity  of  the  material — which  refers  to  the 
eddy-current  losses  in  magnetic  cycles. 
Sp.  gr.  =  specific  gravity  of  the  material. 


FIG.  42. 

Fig.  42  gives  the  magnetic  characteristics,  up  to  H  =  160 
(beyond  this,  the  linear  law  of  reluctivity  usually  applies),  for 
a  number  of  magnetic  materials  of  higher  values  of  saturation 
densities. 

Fig.  43  gives,  with  twice  the  scale  of  ordinates,  but  the  same 


MAGNETISM 


85 


abscissae,  the  magnetic  characteristic  of  some  materials  of  low 
saturation  density. 

Fig.  44  gives,  with  ten  times  the  scale  of  abscissaD,  and  the  same 
scale  of  ordinates,  the  initial  part  of  the  magnetic  characteristic, 


10  20 


40   50   60 


70   80   90   100  110  120  130   140   150 
FIG.  43. 


up  to  H  =  16,  for  the  magnetically  soft  materials  of  Fig.  42, 
that  is,  materials  with  low  value  of  a,  which  rise  so  rapidly  to 
high  values  of  density  that  the  initial  part  of  their  characteristic 
is  not  well  shown  in  the  scale  of  Fig.  42. 

The  magnetic  characteristics  in  Figs.  42,  43  and  44  are  denoted 


86 


ELECTRIC  CIRCUITS 


by  numbers,  and  these  numbers  refer  to  the  materials  given  in  the 
table  of  ' 'Magnetic  Constants"  under  the  same  numbers. 

With  regards  to  the  magnetic  data,  it  must  be  realized,  however, 
that  the  numerical  values,  especially  of  the  less-investigated 
materials,  are  to  some  extent  uncertain,  due  to  the  great  diffi- 
culty of  exact  magnetic  measurements. 


FIG.  44. 

The  saturation  density,  $,  which  is  the  most  constant  and 
most  definite  and  permanent  magnetic  quantity,  can  be  measured 
either  directly,  by  measuring  B  in  such  very  high  fields,—  H  = 
10,000  and  over — that  B—H  does  not  further  increase,  or  in- 
directly, by  observing  the  B,  H  curve  up  to  moderately  high 
fields,  therefrom  derive  the  reluctivity  curve:  p,  H,  and  from  the 
straight-line  law  of  the  latter  curve  determine  <r  and  therewith  B. 


MAGNETISM 


87 


TABLE  I. — SATURATION  DENSITY 
S  =  (B  -  H)H=CO 


Material 

Authority 

Satura- 
tion 
density, 
S 

Demag- 
netization 
tempera- 
ture, 
°C. 

Specific 
gravity 

Iron: 
Most  probable  value  

1916 

X10+3 

21   0-21   5 

760 

7  70 

Best  standard  sheet  steel,  annealed  
Average  standard  sheet  steel,  annealed  
Pure  iron 

1915-16 
1915-16 
Wedekind 

20.70 
20.20 
21    10 

765 

Swedish  wrought  iron  

Ewing 

21.25 

Iron,  99.88  per  cent 

Hatfield 

21   15 

Iron  

Gumlich 

21.60? 

Electrolytic  iron                                       

21.70? 

Commercial  steel 

Williams 

22  00? 

Vacuum-melted  electrolytic  iron  
Pure  iron 

Williams 
DuBois 

22.60? 
23  20? 

756 

7  86 

Average  sheet  iron  
Average  medium  silicon  steel,  annealed,  2.5 
per  cent  

1892 
1915-16 

20.10 
19.25 

Average  soft  steel  castings  

1915-16 

20.20 

20  30 

Magnet  steel  

18.50 

Average  cast  iron 

1915-16 

15.00 

Ft?2Co,  cobalt  iron     

Williams 

22.50? 

520 

Fe2Co,  cobalt  iron,  vacuum-melted  and  forged. 
Fe2Co,  cobalt-iron,  probable  value,  about  

Williams 

25.80? 
23.50 

Scale  of  silicon  steel  

9-10 

Iron  amalgam,  11  per  cent  

1892 

0.90 

Magnetite,  FesCh 

1892 

4.70 

5.10 

Magnetite,  FesO4  

DuBois 

5.46-5.91 

536-589 

Magnetkies,  FerSs  or  FesS» 

DuBois 

0.88 

4.60 

CuFe2O4,  copper  ferrite  

280 

CoFe2O4,  cobalt  ferrite  

280-290 

Cobalt: 
Probable  value  

1916 

12-14 

Cast  cobalt  

11.10 

Cobalt  

12.10 

Cobalt 

Wedekind 

13  30 

Cobalt,  1.66  per  cent.  Fe  
Cobalt           

Ewing 
DuBois 

16.45? 
17.20? 

1075 

8.70 

Cobalt,  pure  

Stifler 

17.85? 

Cobalt,  vacuum-melted  ...            

Williams 

18.85? 

Nickel: 
Probable  value       .  .    

1916 

6-7 

Nickel,  99  per  cent,  pure  

6.15 

Nickel  wire,  soft  

1892 

5.88 

Nickel,  cast  

6.52 

Nickel... 

DuBois 

7.27? 

340-376 

8.93 

Nickel  

8.17? 

Monel  metal  

2.22 

Binel  metal  .         

0.26 

Manganese,  Heusler  alloys: 
AlMnCu2,  soft,  high  permeability  
AlMnCu2,  hard,  high  permeability  

4.67 
3.92 

AlMnCm,  soft,  low  permeability  
AlMnCu2,  highest  values 



1.56 
7.00 

310 

Manganese-antimony,  MnSb  

7.00 

310-330 

Manganese-antimony,  Mn2Sb             

3.85 

Manganese-boron    MnB 

3   10 

Manganese-phosphorus,  MnP  
Manganese-bismuth,  MnB                  



0.70 

18-26 
360-380 

40-50 

Manganese-tin,  MmSn 

Chromium: 
Cr4Og,  chromic  bichromate 
^rjOa,  chromic  chromate 

88 


ELECTRIC  CIRCUITS 


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MAGNETISM 


89 


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90  ELECTRIC  CIRCUITS 

Such  extremely  high  fields,  as  to  reach  complete  magnetic 
saturation,  are  produced  only  between  the  conical  pole  faces  of  a 
very  powerful  large  electromagnet.  The  area  of  the  field  then 
is  very  small,  and  it  is  difficult  to  get  perfect  uniformity  of  the 
field.  The  tendency  is  to  underestimate  the  field,  and  this  gives 
too  high  values  of  S.  Thus,  in  the  following  table  those  values 
of  S,  which  appear  questionable  for  this  case,  have  been  marked 
by  the  interrogation  sign. 

The  indirect  method,  from  the  straight-line  reluctivity  curve, 
gives  more  accurate  values  of  /S,  as  S  is  derived  from  a  complete 
curve  branch,  and  this  method  thus  is  preferable.  However,  the 
value  derived  in  this  manner  is  based  on  the  assumption  that 
there  is  no  further  critical  point  in  the  reluctivity  curve  beyond 
the  observed  range.  This  is  correct  with  iron,  as  the  best  tests 
by  the  direct  method  check.  With  cobalt,  there  may  be  a  critical 
point  in  the  reluctivity  curve  beyond  the  observed  range,  as  there 
are  several  observations  by  the  direct  method,  which  give  very 
much  higher,  though  erratic,  values  of  saturation,  S. 

The  value  of  the  magnetic  hardness,  a,  also  is  difficult  to  de- 
termine for  very  soft  materials,  especially  where  the  method  of 
observation  requires  correction  for  joints,  etc.,  and  the  extremely 
high  values  of  permeability — over  15,000 — therefore  appear 
questionable. 


CHAPTER  VI 

MAGNETISM 

MECHANICAL  FORCES 

1.  General 

51.  Mechanical  forces  appear,  wherever  magnetic  fields  act  on 
electric  currents.  The  work  done  by  all  electric  motors  is  the 
result  of  these  forces.  In  electric  generators,  they  oppose  the 
driving  power  and  thereby  consume  the  power  which  finds  its 
equivalent  in  the  electric  power  output.  The  motions  produced 
by  the  electromagnet  are  due  to  these  forces.  Between  the 
primary  and  the  secondary  coils  of  the  transformer,  between 
conductor  and  return  conductor  of  an  electric  circuit,  etc.,  such 
mechanical  forces  appear. 

The  electromagnet,  and  all  electrodynamic  machinery,  are 
based  on  the  use  of  these  mechanical  forces  between  electric 
conductors  and  magnetic  fields.  So  also  is  that  type  of  trans- 
former which  transforms  constant  alternating  voltage  into  con- 
stant alternating  current.  In  most  other  cases,  however,  these 
mechanical  forces  are  not  used,  and  therefore  are  often  neglected 
in  the  design  of  the  apparatus,  under  the  assumption  that  the 
construction  used  to  withstand  the  ordinary  mechanical  strains 
to  which  the  apparatus  may  be  exposed,  is  sufficiently  strong  to 
withstand  the  magnetic  mechanical  forces.  In  the  large  appara- 
tus, operating  in  the  modern,  huge,  electric  generating  systems, 
these  mechanical  forces  due  to  magnetic  fields  may,  however, 
especially  under  abnormal,  though  not  infrequently  occurring, 
conditions  of  operation  (as  short-circuits),  assume  such  formi- 
dable values,  so  far  beyond  the  normal  mechanical  strains,  as  to  re- 
quire consideration.  Thus  generators  and  large  transformers  on 
big  generating  systems  have  been  torn  to  pieces  by  the  magnetic 
mechanical  forces  of  short-circuits,  cables  have  been  torn  from 
their  supports,  disconnecting  switches  blown  open,  etc. 

In  the  following,  a  general  study  of  these  forces  will  be  given. 
This  also  gives  a  more  rational  and  thereby  more  accurate  de- 

91 


92  ELECTRIC  CIRCUITS 

sign  of  the  electromagnet,  and  permits  the  determination  of  what 
may  be  called  the  efficiency  of  an  electromagnet. 

Investigations  and  calculations  dealing  with  one  form  of 
energy  only,  as  electromagnetic  energy,  or  mechanical  energy, 
usually  are  relatively  simple  and  can  be  carried  out  with  very 
high  accuracy.  Difficulties,  however,  arise  when  the  calculation 
involves  the  relation  between  several  different  forms  of  energy, 
as  electric  energy  and  mechanical  energy.  While  the  elementary 
relations  between  different  forms  of  energy  are  relatively  simple, 
the  calculation  involving  a  transformation  from  one  form  of 
energy  to  another,  usually  becomes  so  complex,  that  it  either  can 
not  be  carried  out  at  all,  or  even  only  approximate  calculation 
becomes  rather  laborious  and  at  the  same  time  gives  only  a  low 
degree  of  accuracy.  In  most  calculations  involving  the  trans- 
formation between  different  forms  of  energy,  it  is  therefore 
preferable  not  to  consider  the  relations  between  the  different 
forms  of  energy  at  all,  but  to  use  the  law  of  conservation  of  energy 
to  relate  the  different  forms  of  energy,  which  are  involved. 

Thus,  when  mechanical  motions  are  produced  by  the  action 
of  a  magnetic  field  on  an  electric  circuit,  energy  is  consumed 
in  the  electric  circuit,  by  an  induced  e.m.f.  At  the  same  time, 
the  stored  magnetic  energy  of  the  system  may  change.  By  the 
law  of  conservation  of  energy,  we  have: 

Electric  energy  consumed  by  the  induced  e.m.f.  =  mechanical 
energy  produced,  +  increase  of  the  stored  magnetic  energy.  (1) 
The  consumed  electric  energy,  and  the  stored  magnetic  energy, 
are  easily  calculated,  as  their  calculation  involves  one  form  of 
energy  only,  and  this  calculation  then  gives  the  mechanical  work 
done,  =  Fl,  where  F  =  mechanical  force,  and  I  =  distance  over 
which  this  force  moves. 

Where  mechanical  work  is  not  required,  but  merely  the  me- 
chanical forces,  which  exist,  as  where  the  system  is  supported 
against  motions  by  the  mechanical  forces — as  primary  and 
secondary  coils  of  a  transformer,  or  cable  and  return  cable  of  a 
circuit — the  same  method  of  calculation  can  be  employed,  by 
assuming  some  distance  I  of  the  motion  (or  dl)\  calculating  the 
mechanical  energy  WQ  =Fl  by  (1),  and  therefrom  the  mechanical 

r  ET        w<>          ET         d"Wo 

force  as  F  =  ^r>  or  F  =  —rr- 
I  at 

Since  the  induced  e.m.f.,  which  consumes  (or  produces)  the 
electric  energy,  and  also  the  stored  magnetic  energy,  depend  on 


MAGNETISM  93 

the  current  and  the  inductance  of  the  electric  circuit,  and  in 
alternating-current  circuits  the  impressed  voltage  also  depends 
on  the  inductance  of  the  circuit,  the  inductance  can  frequently 
be  expressed  by  supply  voltage  and  current;  and  by  substituting 
this  in  equation  (1),  the  mechanical  work  of  the  magnetic  forces 
can  thus  be  expressed,  in  alternating-current  apparatus,  by  sup- 
ply voltage  and  current. 

In  this  manner,  it  becomes  possible,  for  instance,  to  express 
the  mechanical  work  and  thereby  the  pull  of  an  alternating 
electromagnet,  by  simple  expressions  of  voltage  and  current,  or 
to  give  the  mechanical  strains  occurring  in  a  transformer  under 
short-circuits,  by  an  expression  containing  only  the  terminal 
voltage,  the  short-circuit  current,  and  the  distance  between 
primary  and  secondary  coils,  without  entering  into  the  details 
of  the  construction  of  the  apparatus. 

This  general  method,  based  on  the  law  of  conservation  of 
energy,  will  be  illustrated  by  some  examples,  and  the  general 
equations  then  given. 

2.  The  Constant-current  Electromagnet 

52.  Such  magnets  are  most  direct-current  electromagnets, 
and  also  the  series  operating  magnets  of  constant-current  arc 
lamps  on  alternating-current  circuits. 

Let  io  =  current,  which  is  constant  during  the  motion  of 
the  armature  of  the  electromagnet,  from  its  initial  position  1, 
to  its  final  position  2,  I  =  the  length  of  this  motion,  or  the  stroke 
of  the  electromagnet,  in  centimeters,  and  n  =  number  of  turns 
of  the  magnet  winding. 

The  magnetic  flux  $,  and  the  inductance 

(2) 


of  the  magnet,  vary  during  the  motion  of  its  armature,  from  a 
minimum  value, 

$1  =  i^L  i08  (3) 

in  the  initial  position,  to  a  maximum  value, 

$2  =  —  —  108  (4) 

in  the  end  position  of  the  armature. 


94  ELECTRIC  CIRCUITS 

Hereby  an  e.m.f.  is  induced  in  the  magnet  winding, 

,  d$  1r.   ,        .  dL 

e    =  n  -j-  10~8  =  IQ  -r-  (5) 

This  consumes  the  power 
and  thereby  the  energy 


=    P 


w  =        pdt  =  m02(L2  -  Li)  (7) 


Assuming  that  the  inductance,  in  any  fixed  position  of  the 
armature,  does  not  vary  with  the  current,  that  is,  that  magnetic 
saturation  is  absent,1  the  stored  magnetic  energy  is: 

In  the  initial  position,  1, 

IQ  LI  ,_.* 

MI  =  — 2~  (8) 

in  the  end  position,  2, 

^02-£/2  ,„>. 

^2  =  —3—  (9) 

The  increase  of  the  stored  magnetic  energy,  during  the  motion 
of  the  armature,  thus  is 

w'  =  w2  —  Wi  =  ^-  (L2  —  LI)  (10) 

The  mechanical  work  done  by  the  electromagnet  thus  is, 
by  the  law  of  conservation  of  energy, 

WQ   =   W  —  Wf 

=  -^-  (L2  —  LI)  joules.  (11) 

If  I  =  length  of  stroke,  in  centimeters,  F  =  average  force, 
or  pull  of  the  magnet,  in  gram  weight,  the  mechanical  work  is 

Fl  gram-cm. 
Since 

g  =  981  cm.-sec.  (12) 

=  acceleration  of  gravity,  the  mechanical  work  is,  in  absolute 
units, 

Fig 

1  If  magnetic  saturation  is  reached,  the  stored  magnetic  energy  is  taken 
from  the  magnetization  curve,  as  the  area  between  this  curve  and  the 
vertical  axis,  as  discussed  before. 


MAGNETISM  95 

and  since  1  joule  =  107  absolute  units,  the  mechanical  work  is 
w0  =  Fig  10+7  joules.  (13) 

From  (11)  and  (12)  then  follows, 

Fl  =  ^  (L2  -  L010+7  gram-cm.  (14) 

*Q 

as  the  mechanical  work  of  the  electromagnet,  and 

F-g^j^i  10'  grams  (15) 

as  the  average  force,  or  pull  of  the  electromagnet,  during  its 
stroke  I. 

Or,  if  we  consider  only  a  motion  element  dl, 


=    g  gramS 

as  the  force,  or  pull  of  the  electromagnet  in  any  position  I. 

Reducing  from  gram-centimeters  to  foot-pounds,  that  is,  giving 
the  stroke  I  in  feet,  the  pull  F  in  pounds,  we  divide  by 

454  X  30.5  =  13,850 
which  gives,  after  substituting  for  g  from  (12) 

(14)  :  Fl  =  3.68  i0»(Z,a  -  Li)  ft.-lb  (17) 

(15)  :   F  =  3.68  i<?  L*  ~  Ll  Ib.  (18) 

(16):   F  =  3.68zV^lb.  (19) 

These  equations  apply  to  the  direct-current  electromagnet 
as  well  as  to  the  alternatingTcurrent  electromagnet. 

In  the  alternating-current  electromagnet,  if  i0  is  the  effective 
value  of  the  current,  F  is  the  effective  or  average  value  of  the 
pull,  and  the  pull  or  force  of  the  electromagnet  pulsates  with 
double  frequency  between  0  and  2  F. 

53.  In  the  alternating-current  electromagnet  usually  the  vol- 
tage consumed  by  the  resistance  of  the  winding,  t'0r,  can  be 
neglected  compared  with  the  voltage  consumed  by  the  reactance 
of  the  winding,  i0x,  and  the  latter,  therefore,  is  practically  equal 
to  the  terminal  voltage,  e,  of  the  electromagnet.  We  have  then, 
by  the  general  equation  of  self-induction, 

e  =  2rr  fLi0  (20) 


96  ELECTRIC  CIRCUITS 

where  /  =  frequency,  in  cycles  per  second. 
From  which  follows, 


ioL  =  ...         (21) 

and  substituting  (21)  in  equations  (14)  to  (19),  gives  as  the  equa- 
tion of  the  mechanical  work,  and  the  pull  of  the  alternating-current 
electromagnet. 

In  the  metric  system: 

-7  gram-cm.  '  (22) 

(23) 


In  foot-pounds: 

pl  =  0.586  fo(*  -  ej  ft>_lb>  (24) 

_  0.586  iQ(e2  -  ej  _  0.586?'0  de 

fl  f       dl  !t 

Example.  —  In  a  60-cycle  alternating-current  lamp  magnet, 
the  stroke  is  3  cm.,  the  voltage,  consumed  at  the  constant  alter- 
nating current  of  3  amp.  is  8  volts  in  the  initial  position,  17 
volts  in  the  end  position.  What  is  the  average  pull  of  the 
magnet? 

I    =3  cm. 
el  =  8 
e2  =  17 
/    =  60 
io   =  3 
hence,  by  (23), 

F  =  122  grams  (=  0.27  Ib.) 

The  work  done  by  an  electromagnet,  and  thus  its  pull,  depend, 
by  equation  (22),  on  the  current  io  and  the  difference  in  voltage 
between  the  initial  and  the  end  position  of  the  armature,  ez  —  e\', 
that  is,  depend  upon  the  difference  in  the  volt-amperes  con- 
sumed by  the  electromagnet  at  the  beginning  and  at  the  end  of  the 
stroke.  With  a  given  maximum  volt-amperes,  z'o62,  available 
for  the  electromagnet,  the  maximum  work  would  thus  be  done, 
that  is,  the  greatest  pull  produced,  if  the  volt-amperes  at  the 
beginning  of  the  stroke  were  zero,  that  is,  e\  =  0,  and  the 
theoretical  maximum  output  of  the  magnet  thus  would  be 

(26) 


MAGNETISM  97 

and  the  ratio  of  the  actual  output,  to  the  theoretically  maximum 
output,  or  the  efficiency  of  the  electromagnet,  thus  is,  by  (22) 
and  (26)  > 


—    rr 

/'m  #2 

or,  using  the  more  general  equation  (14),  which  also  applies  to 
the  direct-current  electromagnet, 


-L/2    — 


/no\ 

(28) 


The  efficiency  of  the  electromagnet,  therefore,  is  the  dif- 
ference between  maximum  and  minimum  voltage,  divided  by  the 
maximum  voltage;  or  the  difference  between  maximum  and 
minimum  volt-ampere  consumption,  divided  by  the  maximum 
volt-ampere  consumption;  or  the  difference  between  maximum 
and  minimum  inductance,  divided  by  the  maximum  inductance. 

As  seen,  this  expression  of  efficiency  is  of  the  same  form  as 
that  of  the  thermodynamic  engine, 


From  (26)  it  also  follows,  that  the  maximum  work  which  can 
be  derived  from  a  given  expenditure  of  volt-amperes,  ioe2,  is 
limited.  For  i0ez  =  1,  that  is,  for  1  volt-amp,  the  maximum 
work,  which  could  be  derived  from  an  alternating  electro- 
magnet, is,  from  (26), 

„  ,        107        810  ,00, 

Fml  =  j— j-  =  -j-  gram-cm.  (29) 

That  is,  a  60-cycle  electromagnet  can  never  give  more  than 
13.5  gram-cm.,  and  a  25-cycle  electromagnet  never  more  than 
32.4  gram-cm,  pull  per  volt-ampere  supplied  to  its  terminals. 

Or  inversely,  for  an  average  pull  of  1  gram  over  a  distance  of 

1  cm.,  a  minimum  of  y~-^  volt-amp,  is  required  at  60  cycles, 

and  a  minimum  of  x^f  volt-amp,  at  25  cycles. 

Or,  reduced  to  pounds  and  inches: 

For  an  average  pull  of  1  Ib.  over  a  distance  of  1  in.,  at  least 
86  volt-amp,  are  required  at  60  cycles,  and  at  least  36  volt- 
amp,  at  25  cycles. 

This  gives  a  criterion  by  which  to  judge  the  success  of  the 
design  of  electromagnets. 

7 


98  ELECTRIC  CIRCUITS 

3.  The  Constant-potential  Alternating  Electromagnet 

54.  If  a  constant  alternating  potential,  e0,  is  impressed  upon  an 
electromagnet,  and  the  voltage  consumed  by  the  resistance, 
ir,  can  be  neglected,  the  voltage  consumed  by  the  reactance,  x,  is 
constant  and  is  the  terminal  voltage,  eQ,  thus  the  magnetic  flux, 
<&,  also  is  constant  during  the  motion  of  the  armature  of  the 
electromagnet.  The  current,  i,  however,  varies,  and  decreases 
from  a  maximum,  ii,  in  the  initial  position,  to  a  minimum,  z'2,  in 
the  end  position  of  the  armature,  while  the  inductance  increases 
from  Z/i  to  L 2. 

The  voltage  induced  in  the  electric  circuit  by  the  motion  of 

the  armature, 

/7d> 

e'  =  n~  108  (30) 

at 

then  is  zero,  and  therefore  also  the  electrical  energy  expended, 

w  =  0. 

That  is,  the  electric  circuit  does  no  work,  but  the  mechanical 
work  of  moving  the  armature  is  done  by  the  stored  magnetic 
energy. 

The  increase  of  the  stored  magnetic  energy  is 

(31) 


2 

and  since  the  mechanical  energy,  in  joules,  is  by  (13), 

WQ  =  Fig  107 
the  equation  of  the  law  of  conservation  of  energy, 

w  =  wf  +  WQ  (32) 

then  becomes 


o 

or 

Fl  =  ll*Ll  ~  l^L<i  107  gram-cm.  (33) 

Since,  from  the  equation  of  self-induction,  in  the  initial  posi- 
tion, 

e0  =  2  Tr/Lit!  (34) 

in  the  end  position 

(35) 


MAGNETISM  99 

substituting  (34)  and  (35)  in   (33),  gives  the  equation  of  the 
constant-potential  alternating  electromagnet. 

Fl  -=  ^^  1°7  gram-cm.  (36) 

and 

p  =  <^H^  107  =  -^  %  107  grams  (37) 

4wfgl  47T/0  dl 

or,  in  foot-pounds, 

0.586  <-*2) 


0.586  cody  -  i2)       0.586  e0  di 

F  ~JT  ~J~~  Jl[b' 

Substituting  Q  =  ei  =  volt-amperes,  in  equations  (36)  to 
(39)  of  the  constant-potential  alternating  electromagnet,  and 
equations  (22)  to  (25)  of  the  constant-current  alternating  magnet, 
gives  the  same  expression  of  mechanical  work  and  pull  : 

In  metric  system: 

Fl  =  ^-  107  gram-cm.  (40) 

47T/0 

F  =  -£%-.  107  =  -pV  ^  107  grams  (41) 

4  irfgl  4  irfg  dl 

In  foot-pounds  : 

pl  =  0.586  AQft_lb  (42) 

,,       0.586  AQ       0.586  dQ 
F         ~^T         ~J~  ~dl 

where  AQ  =  difference  in  volt-amperes  consumed  by  the  magnet 
in  the  initial  position,  and  in  the  end  position  of  the  armature. 

Both  types  of  alternating-current  magnet,  then,  give  the  same 
expression  of  efficiency, 

"g    • 

where  Qm  is  the  maximum  volt-amperes  consumed,  corresponding 
to  the  end  position  in  the  constant-current  magnet,  to  the  initial 
position  in  the  constant-potential  magnet. 

4.  Short-circuit  Stresses  in  Alternating-current  Transformers 

55.  At  short-circuit,  no  magnetic  flux  passes  through  the  sec- 
ondary coils  of  the  transformer,  if  we  neglect  the  small  voltage 
consumed  by  the  ohmic  resistance  of  the  secondary  coils.  If 


100  ELECTRIC  CIRCUITS 

the  supply  system  is  sufficiently  large  to  maintain  constant 
voltage  at  the  primary  terminals  of  the  transformer  even  at 
short-circuit,  full  magnetic  flux  passes  through  the  primary 
coils.1  In  this  case  the  total  magnetic  flux  passes  between 
primary  coils  and  secondary  coils,  as  self-inductive  or  leakage 
flux.  If  then  x  =  self-inductive  or  leakage  reactance,  eQ  =  im- 
pressed e.m.f.,  IQ  =  —  is  the  short-circuit  current  of  the  trans- 
*c 

former.  Or,  if  as  usual  the  reactance  is  given  in  per  cent.,  that 
is,  the  ix  (where  i  —  full-load  current  of  the  transformer)  given 
in  per  cent,  of  e,  the  short-circuit  current  is  equal  to  the  full-load 
current  divided  by  the  percentage  reactance.  Thus  a  trans- 
former with  4  per  cent,  reactance  would  give  a  short-circuit  cur- 
rent, at  maintained  supply  voltage,  of  25  times  full-load  current. 

To  calculate  the  force,  F,  exerted  by  this  magnetic  leakage 
flux  on  the  transformer  coils  (which  is  repulsion,  since  primary 
and  secondary  currents  flow  in  opposite  direction)  we  may  assume, 
at  constant  short-circuit  current,  io,  the  secondary  coils  moved 
against  this  force,  F,  and  until  their  magnetic  centers  coincide 
with  those  of  the  primary  coils;  that  is,  by  the  distance,  I,  as  shown 
diagrammatically  in  Fig.  45,  the  section  of  a  shell-type  transformer. 
When  brought  to  coincidence,  no  magnetic  flux  passes  between 
primary  and  secondary  coils,  and  during  this  motion,  of  length,  I, 
the  primary  coils  thus  have  cut  the  total  magnetic  flux,  <f>,  of  the 
transformer. 

Hereby  in  the  primary  coils  a  voltage  has  been  induced, 

6'  =  n  ^  10-8 
at 

where  n  =  effective  number  of  primary  turns. 

The  work  done  or  rather  absorbed  by  this  voltage,  e',  at  cur- 
rent, z'o,  is 

w  =    I  e'iodt  =  m'o^lO"8  joules.  (45) 

1  If  the  terminal  voltage  drops  at  short-circuit  on  the  transformer  seconda- 
ries, the  magnetic  flux  through  the  transformer  primaries  drops  in  the  same 
proportion,  and  the  mechanical  forces  in  the  transformer  drop  with  the 
square  of  the  primary  terminal  voltage,  and  with  a  great  drop  of  the  ter- 
minal voltage,  as  occurs  for  instance  with  large  transformers  at  the  end  of 
a  transmission  line  or  long  feeders,  the  mechanical  forces  may  drop  to  a 
small  fraction  of  the  value,  which  they  have  on  a  system  of  practically  un- 
limited power. 


MAGNETISM  101 


If  L  =  leakage  inductance  of  the  transformer,  at  ghort-ckeiiit, 
where  the  entire  flux,  <£,  is  leakage  flux,  ^e  haVe:     i  ;  •  J;%  :  ',  ;,  ;  * 

$  =  —  °108  '(46) 

n 

hence,  substituted  in  (45) 

w  =  i0zL  (47) 

The  stored  magnetic  energy  at  short-circuit  is 

»i-if!  (48) 

and  since  at  the  end  of  the  assumed  motion  through  distance,  Z, 
the  leakage  flux  has  vanished  by  coincidence  between  primary 
and  secondary  coils,  its  stored  magnetic  energy  also  has  vanished, 
and  the  change  of  stored  magnetic  energy  therefore  is 

w'  =  w,  =  *f  (49) 

Hence,  the  mechanical  work  of  the  magnetic  forces  of  the  short- 
circuit  current  is 

Wo  =  w  -  w'  =  ^  (50) 

It  is,  however,  if  F  is  the  force,  in  grams,  /,  the  distance  between 
the  magnetic  centers  of  primary  and  secondary  coils, 

wi  =  Fig  10-7  joules. 
Hence, 

Fl  =  ^  107  gram-cm.  (51) 

^  9 
and 


F  =          10'  grams  (52) 

the  mechanical  force  existing  between  primary  and  secondary 
coils  of  a  transformer  at  the  short-circuit  current,  iQ. 

Since  at  short-circuit,  the  total  supply  voltage,  eQ,  is  consumed 
by  the  leakage  inductance  of  the  transformer,  we  have 

e0  =  2  wfLio  (53) 

hence,  substituting  (53)  in  (52),  gives 

107 


grams  (54) 


102  ELECTRIC  CIRCUITS 

Example.  —  Let,  in  a  25-cycle  1667-kw.  transformer,  the  supply 
voltage,  c0  —  -520(},-.the  reactance  =  4  per  cent.  The  trans- 
former contains  two  primary  coils  between  three  secondary  coils, 
and  the  distance  between  the  magnetic  centers  of  the  adjacent 
coils  or  half  coils  is  12  cm.,  as  shown  diagrammatically  in  Fig.  45. 
What  force  is  exerted  on  each  coil  face  during  short-circuit,  in  a 
system  which  is  so  large  as  to  maintain  constant  terminal  voltage? 

At  5200  volts  and  1667  kw.,  the  full-load  current  is  320  amp. 
At  4  per  cent,  reactance  the  short-circuit  current  therefore, 

QOn 

io  =  ~-~j  =  8000  amp.     Equation  (54)  then  gives,  for  /  =  25, 

I  =  12, 

F  =  112  X  IO6  grams 
=  112  tons. 

This  force  is  exerted  between  the  four  faces  of  the  two  primary 
coils,  and  the  corresponding  faces  of  the  secondary  coils,  and 
on  every  coil  face  thus  is  exerted  the  force 

^  =  28  tons 

This  is  the  average  force,   and  the  force  varies  with  double 
frequency,  between  0  and  56  tons,  and  is  thus  a  large  force. 

56.  Substituting  iQ  =  —  in  (54),  gives  as  the  short-circuit  force 
x 

of  an  alternating-current  transformer,  at  maintained  terminal 
voltage,  e0,  the  value 

,,       e02  IO7       810  e02 

F  -  -  "JET  grams  (55) 


That  is,  the  short-circuit  stresses  are  inversely  proportional 
to  the  leakage  reactance  of  the  transformer,  and  to  the  distance, 
I,  between  the  coils. 

In  large  transformers  on  systems  of  very  large  power,  safety 
therefore  requires  the  use  of  as  high  reactance  as  possible. 

High  reactance  is  produced  by  massing  the  coils  of  each  cir- 
cuit. 

Let  in  a  transformer 

n    =  number  of  coil  groups 


MAGNETISM 


103 


(where  one  coil  is  divided  into  two  half  coils,  one  at  each  end  of 
the  coil  stack,  as  one  secondary  coil  in  Fig.  45j  where  n  =  2)  the 
mechanical  force  per  coil  face  then  is,  by  (55), 

F         e02  107         810  e02 
Fo  =  2n  =  8^gnTx==  2/to  gramS 
Let  x  =  leakage  reactance  of  transformer; 
1Q  =  distance  between  coil  surfaces; 

1 1  =  thickness  of  primary  coil; 

12  =  thickness  of  secondary  coil. 

Between  two  adjacent  coils,  P  and  S  in  Fig.  45,  the  leakage  flux 
density  is  uniform  for  the  width  10  between  the  coil  surfaces, 


1           CO 

co        4 

£ 

F 

TJ 

V 

-o 

1 

1*. 

* 

CO 

t 

CO 

t 

1*, 

[», 

1 
•o 

t 

•o 

f 

[•s 

*             CO 

CO            4 

1 

FIG.  45. 

and  then  decreases  toward  the  interior  of  the  coils,  over  the  dis- 
tance ^  respectively  ^,  to  zero  at  the  coil  centers.  All  the  coil 

turns  are  interlinked  with  the  leakage  flux  in  the  width,  10)  but 
toward  the  interior  of  the  coils,  the  number  of  turns  interlinked 
with  the  leakage  flux  decreases,  to  zero  at  the  coil  center,  and  as 
the  leakage  flux  density  also  decreases,  proportional  to  the  dis- 
tance from  the  coil  center,  to  zero  in  the  coil  center,  the  inter- 
linkages  between  leakage  flux  and  coil  turns  decrease  over  the 

space  2  respectively  ^ ,  proportional  to  the  square  of  the  distance 
from  the  coil  center,  thus  giving  a  total  interlinkage  distance, 


ll 

=  6' 


where  u  is  the  distance  from  the  coil  center. 


104  ELECTRIC  CIRCUITS 

Thus  the  total  interlinkages  of  the  leakage  flux  with  the  coil 
turns  are  the  same  as  that  of  a  uniform  leakage  flux  density  over 

the  width  10  +  ^-  +  ^-     This  gives  the  effective  distance  between 
coil  centers,  for  the  reactance  calculation, 

l-h  +  l+±±  (57) 

Assuming  now  we  regroup  the  transformer  coils,  so  as  to  get 
m  primary  and  m  secondary  coils,  leaving,  however,  the  same  iron 
structure. 

The  leakage  flux  density  between  the  coils  is  hereby  changed  in 
proportion  to  the  changed  number  of  ampere-turns  per  coil,  that 

is,  by  the  factor  — 

The  effective  distance  between  the  coils,  I,  is  changed  by  the 

same  factor  —  • 

m 

The  number  of  interlinkages  between  leakage  flux  and  electric 
circuits,  and  thus  the  leakage  reactance,  x}  of  the  transformer, 
thus  is  changed  by  the  factor 


That  is,  by  regrouping  the  transformer  winding  within  the  same 
magnetic  circuit  and  without  changing  the  number  of  turns  of  the 
electric  circuit,  the  leakage  reactance,  x,  changes  inverse  propor- 
tional to  the  square  of  the  number  of  coil  groups. 

As  by  equation  (56)  the  mechanical  force  is  inverse  propor- 

(H\  ^ 
—  )   >  I  pro- 

portional to  —  »   the   mechanical   force    per    coil    thus    changes 
proportional  to 


x  -  x  - 
m      X  m  X   n         m 


That  is,  regrouping  the  transformer  winding  in  the  same  wind- 
ing space  changes  the  mechanical  force  inverse  proportional  to 


MAGNETISM  105 

the  square  of  the  coil  groups,  thus  inverse  proportional  to  the 
change  of  leakage  reactance. 

However,  the  distance  1Q  between  the  coils  is  determined  by  in- 
sulation and  ventilation.  Thus  its  decrease,  when  increasing  the 
number  of  coil  groups,  would  usually  not  be  permissible,  but  more 
winding  space  would  have  to  be  provided  by  changing  the  mag- 
netic circuit,  and  inversely,  with  a  reduction  of  the  number  of 
coil  groups,  the  winding  space,  and  with  it  the  magnetic  circuit, 
would  be  reduced. 

Assuming,  then,  that  at  the  change  from  n  to  m  coil  groups, 
the  distance  between  the  coils,  IQ,  is  left  the  same. 

The  effective  leakage  space  then  changes  from 


to 


,    n  h  + 
c 


i/  _  7   _L_  n  *'  +  ^  _  7   ^    '   m       6 

'     —   &0     i     —  7; —   t  7~ — ; — 5 — > 

m       6  .     ,    Zi  +  12 


and  the  leakage  reactance  thus  changes  from 

x 
to 

,       n  lf    . 
x    =  --  -rx> 
m  t 

hence  the  mechanical  force  per  coil,  from 

F          e0*  107 


2n       8  irfnglx 
to 

F'          eo2  107 


2m      8  ifngl'x* 

„     nix 
0  ml'x' 


(F) 


(58) 


106  ELECTRIC  CIRCUITS 

Thus,  if  — ~ —  is  large  compared  with  Z0, 


*'-©'*. 


that  is,  the  mechanical  forces  vary  with  the  square  of  the  number 
of  coil  groups. 

If  is  small  compared  with  Z0, 


that  is,  the  mechanical  forces  are  not  changed  by  the  change  of 
the  number  of  coil  groups. 

In  actual  design,  decreasing  the  number  of  coil  groups  usually 
materially  decreases  the  mechanical  forces,  but  materially  less 
than  proportional  to  the  square  of  the  number  of  coil  groups. 

5.  Repulsion  between  Conductor  and  Return  Conductor 

57.  If  IQ  is  the  current  flowing  in  a  circuit  consisting  of  a  con- 
ductor and  the  return  conductor  parallel  thereto,  and  I  the  dis- 
tance between  the  conductors,  the  two  conductors  repel  each 
other  by  the  mechanical  force  exerted  by  the  magnetic  field  of 
the  circuit,  on  the  current  in  the  conductor. 

As  this  case  corresponds  to  that  considered  in  section  2,  equa- 
tion (16)  applies,  that  is, 


The  inductance  of  two  parallel  conductors,  at  distance  I  from 
each  other,  and  conductor  diameter  ld  is,  per  centimeter  length  of 
conductor, 

L  =   (4  log  ~  +  M)  10-9  henrys  (59) 

Hence,  differentiated, 

dL  =  4  X  IP"9 
dl  =  I 

and,  substituted  in  (16), 


or  substituting  (12), 


MAGNETISM  107 

20.4  ?02  10~6 
F  =  -     -j—     -  grams  (61) 

If  I  =  150  cm.  (5  ft.) 

io  =  200  amp. 
this  gives 

F  =  0.0054  grams  per  centimeter  length  of  circuit,  hence  it  is 
inappreciable. 

If,  however,  the  conductors  are  close  together,  and  the  current 
very  large,  as  the  momentary  short-circuit  current  of  a  large 
alternator,  the  forces  may  become  appreciable. 

For  example,  a  2200-volt  4000-kw.  quarter-phase  alternator 
feeds  through  single  conductor  cables  having  a  distance  of  15  cm. 
(6  in.)  from  each  other.  A  short-circuit  occurs  in  the  cables,  and 
the  momentary  short-circuit  current  is  12  times  full-load  current. 
What  is  the  repulsion  between  the  cables? 

Full-load  current  is,  per  phase,  910  amp.  Hence,  short-circuit 
current,  i0  =  12  X  910  =  10,900  amp.  I  =  15.  Hence, 

F  =  160  grams  per  centimeter. 
Or  multiplied  by 


F  =  10.8  Ib.  per  feet  of  cable. 

That  is,  pulsating  between  0  and  21.6  Ib.  per  foot  of  cable. 
Hence  sufficient  to  lift  the  cable  from  its  supports  and  throw  it 
aside. 

In  the  same  manner,  similar  problems,  as  the  opening  of  dis- 
connecting switches  under  short-circuit,  etc.,  can  be  investigated. 

6.  General  Equations  of  Mechanical  Forces  in  Magnetic  Fields 

58.  In  general,  in  an  electromagnetic  system  in  which  mechan- 
ical motions  occur,  the  inductance,  L,  is  a  function  of  the  position,  /, 
during  the  motion.  If  the  system  contains  magnetic  material, 
in  general  the  inductance,  L,  also  is  a  function  of  the  current,  i, 
especially  if  saturation  is  reached  in  the  magnetic  material. 

Let,  then,L  =  inductance,  as  function  of  the  current,  iy  and 
position,  I', 

LI  =  inductance,  as  function  of  the  current,  i,  in  the  initial 
position  1  of  the  system; 

L2  =  inductance,  as  function  of  the  current,  i,  in  the  end 
position  2  of  the  system. 


108  ELECTRIC  CIRCUITS 

If  then  <J>  =  magnetic  flux,  n  =  number  of  turns  interlinked 
with  the  flux,  the  induced  e.m.f.  is 

f/<J> 
e'  =  n  ^  10-8  (62) 

We  have,  however, 

n$  =  iL  108; 
hence, 

•  (63) 


the  power  of  this  induced  e.m.f.  is 

.  d(iL) 

p  =  ie    =  i    ^     ' 

dt 
and  the  energy 

w 


=   I    pdt  =   I    id(iL) 

=   rpdL  +  FiLdi  (64) 

The  stored  magnetic  energy  in  the  initial  position  1  is 

wi  =   f  id(iLi)  (65) 

Jo 

In  the  end  position  2, 

2  (66) 


o 

and  the  mechanical  work  thus  is,  by  the  law  of  conservation  of 
energy 

WQ   =   W  —  Wz  +   Wi 

=    f2id(iL)  +   f  id(iLi)  -    C  id(iL2)  (67) 

Ji  Jo  Jo 

and  since  the  mechanical  work  is 

wo  =  Fig  10-7  (68) 

We  have  : 

1Q7    f     /»2  /»!  /»2  ] 

Fl  =  —  {       id(iL)  +  I    id(iLi)  -   I    id(iL*)     gram-cm.     (69) 
9    [Ji  Jo  Jo 


MAGNETISM  109 

If  L  is  not  a  function  of  the  current,  i,  but  only  of  the  position, 
that  is,  if  saturation  is  absent,  LI  and  Z/2  are  constant,  and  equa- 
tion (69)  becomes, 


J 


id(iL}  +  -  ^       *2     l  \  gram-cm.         (70) 


(a)  If  t  =  constant,  equation  (70)  becomes, 


_  10_7  i*(L2  -Li) 
<7  2 

(Constant-current  electromagnet.) 
(6)  If  L  =  constant,  equation  (70)  becomes, 

Fl  =  0. 

That  is,  mechanical  forces  are  exerted  only  where  the  in- 
ductance of  the  circuit  changes  with  the  mechanical  motion 
which  would  be  produced  by  these  forces. 

(c)  If  iL  —  constant,  equation  (70)  becomes, 

107  iL(ii  -  it) 
T       ~^~ 
(Constant-potential  electromagnet.) 

In  the  general  case,  the  evaluation  of  equation  (69)  can  usually 
be  made  graphically,  from  the  two  curves,  which  give  the  varia- 
tion of  Z/i  with  i  in  the  initial  position,  of  L2  .with  i  in  the  final 
position,  and  the  curve  giving  the  variation  of  L  and  i  with  the 
motion  from  the  initial  to  the  final  position. 

In  alternating  magnetic  systems,  these  three  curves  can  be 
determined  experimentally  by  measuring  the  volts  as  function 
of  the  amperes,  in  the  fixed  initial  and  end  position,  and  by 
measuring  volts  and  amperes,  as  function  of  the  intermediary 
positions,  that  is,  by  strictly  electrical  measurement. 

As  seen,  however,  the  problem  is  not  entirely  determined  by 
the  two  end  positions,  but  the  function  by  which  i  and  L  are 
related  to  each  other  in  the  intermediate  positions,  must  also  be 
given.  That  is,  in  the  general  case,  the  mechanical  work  and 
thus  the  average  mechanical  force,  are  not  determined  by  the 
end  positions  of  the  electromagnetic  system.  This  again  shows 
an  analogy  to  thermodynamic  relations. 

If  then  in  case  of  a  cyclic  change,  the  variation  from  position 


110  ELECTRIC  CIRCUITS 

1  to  2  is  different  from  that  from  position  2  back  to  1,  such  a 
cyclic  change  produces  or  consumes  energy. 


w 


=    I  id(iL)  +   I  id(iL)  =   \    id(iL) 


Such  a  case  is  the  hysteresis  cycle.  The  reaction  machine 
(see  Theory  and  Calculation  of  Electrical  Apparatus)  is  based 
on  such  cycle. 


SECTION  II 

CHAPTER  VII 
SHAPING  OF  WAVES :  GENERAL 

59.  In  alternating-current  engineering,  the  sine  wave,  as  shown 
in  Fig.  46,  is  usually  aimed  at  as  the  standard.  This  is  not  due  to 
any  inherent  merit  of  the  sine  wave. 

For  all  those  purposes,  where  the  energy  developed  by  the  cur- 
rent in  a  resistance  is  the  object,  as  for  incandescent  lighting, 
heating,  etc.,  any  wave  form  is  equally  satisfactory,  as  the  energy 
of  the  wave  depends  only  on  its  effective  value,  but  not  on  its 
shape. 

With  regards  to  insulation  stress,  as  in  high-voltage  systems,  a 
flat-top  wave  of  voltage  and  current,  such  as  shown  in  Fig.  47, 
would  be  preferable,  as  it  has  a  higher  effective  value,  with  the 
same  maximum  value  and  therefore  with  the  same  strain  on  the 
insulation,  and  therefore  transmits  more  energy  than  the  sine 
wave,  Fig.  46. 

Inversely,  a  peaked  wave  of  voltage,  such  as  Fig.  48,  and  such 
as  the  common  saw-tooth  wave  of  the  unitooth  alternator,  is 
superior  in  transformers  and  similar  devices,  as  it  transforms  the 
energy  with  less  hysteresis  loss.  The  peaked  voltage  wave,  Fig. 
48,  gives  a  flat-topped  wave  of  magnetism,  Fig.  47,  and  thereby 
transforms  the  voltage  with  a  lesser  maximum  magnetic  flux,  than 
a  sine  wave  of  the  same  effective  value,  that  is,  the  same  power. 
As  the  hysteresis  loss  depends  on  the  maximum  value  of  the  mag- 
netic flux,  the  reduction  of  the  maximum  value  of  the  magnetic 
flux,  due  to  a  peaked  voltage  wave,  results  in  a  lower  hysteresis 
loss,  and  thus  higher  efficiency  of  transformation.  This  reduc- 
tion of  loss  may  amount  to  as  much  as  15  to  25  per  cent,  of  the 
total  hysteresis  loss,  in  extreme  cases. 

Inversely,  a  peaked  voltage  wave  like  Fig.  48  would  be  objec- 
tionable in  high-voltage  transmission  apparatus,  by  giving  an  un- 
necessary high  insulation  strain,  and  a  flat-top  wave  of  voltage 
like  Fig.  47,  when  impressed  upon  a  transformer,  would  give  a 
peaked  wave  of  magnetism  and  thereby  an  increased  hysteresis 
loss. 

Ill 


112 


ELECTRIC  CIRCUITS 


The  advantage  of  the  sine  wave  is,  that  it  remains  unchanged  in 
shape  under  most  conditions,  while  this  is  not  the  case  with  any 
other  wave  shape,  and  any  other  wave  shape  thus  introduces  the 
danger,  that  under  certain  conditions,  or  in  certain  parts  of  the 
circuit,  it  may  change  to  a  shape  which  is  undesirable  or  even 


Fia.  46. 


/Fia.  47. 


"\ 


Fia.  48. 


V 

Fia.  49. 


FIGS.  46  TO  49. 

dangerous.  Voltage,  e,  and  current,  i,  are  related  to  each  other  by 
proportionality,  by  differentiation  and  by  integration,  with  re- 
sistance, r,  inductance,  L,  and  capacity,  C,  as  factors, 

e  =  ri, 


=  C  fidt, 


and  as  the  differentials  and  integrals  of  sines  are  sines,  as  long  as 
r,  L  and  C  are  constant — which  is  mostly  the  case — sine  waves  of 


SHAPING  OF  WAVES 


113 


voltage  produce  sine  waves  of  current  and  inversely,  that  is,  the 
sine  wave  shape  of  the  electrical  quantities  remains  constant. 

A  flat-topped  current  wave  like  Fig.  47,  however,  would  by 
differentiation  give  a  self-inductive  voltage  wave,  which  is  peaked, 
like  Fig.  48.  A  voltage  wave  like  Fig.  48,  which  is  more  efficient 
in  transformation,  may  by  further  distortion,  as  by  intensifica- 
tion of  the  triple  harmonic  by  line  capacity,  assume  the  shape, 


FIG.  50. 

Fig.  49,  and  the  latter  then  would  give,  when  impressed  upon  a 
transformer,  a  double-peaked  wave  of  magnetism,  Fig.  50,  and 
such  wave  of  magnetism  gives  a  magnetic  cycle  with  two  small 


FIG.  51. 

secondary  loops  at  high  density,  as  shown  in  Fig.  51,  and  an 
additional  energy  loss  by  hysteresis  in  these  two  secondary  loops, 
which  is  considerable  due  to  the  high  mean  magnetic  density,  at 
which  the  secondary  loop  is  traversed,  so  that  in  spite  of  the 
reduced  maximum  flux  density,  the  hysteresis  loss  may  be 
increased. 

Therefore,  in  alternating-current  engineering,  the  aim  gener- 

8 


114  ELECTRIC  CIRCUITS 

ally  is  to  produce  and  use  a  wave  which  is  a  sine  wave  or 
nearly  so. 

60.  In  an  alternating-current  generator,  synchronous  or  in- 
duction machine,  commutating  machine,  etc.,  the  wave  of  voltage 
induced  in  a  single  armature  conductor  or  "face  conductor" 
equals  the  wave  of  field  flux  distribution  around  the  periphery  of 
the  magnet  field,  modified,  however,  by  the  reluctance  pulsations 
of  the  magnetic  circuit,  where  such  exist.  As  the  latter  produce 
higher  harmonics,  they  are  in  general  objectionable  and  to  be 
avoided  as  far  as  possible. 

By  properly  selecting  the  length  of  the  pole  arc  and  the  length 
of  the  air-gap  between  field  and  armature,  a  sinusoidal  field  flux 
distribution  and  thereby  a  sine  wave  of  voltage  induced  in  the 
armature  face  conductor  could  be  produced.  In  this  direction, 
however,  the  designer  is  very  greatly  limited  by  economic  con- 
sideration: length  of  pole  arc,  gap  length,  etc.,  are  determined 
within  narrow  limits  by  the  requirement  of  the  economic  use  of 
the  material,  questions  of  commutation,  of  pole-face  losses,  of  field 
excitation,  etc.,  so  that  as  a  rule  the  field  flux  distribution  and 
with  it  the  voltage  induced  in  a  face  conductor  differs  materially 
from  sine  shape. 

The  voltage  induced  in  a  face  conductor  may  contain  even  har- 
monics as  well  as  odd  harmonics,  and  often,  as  in  most  inductor 
alternators,  a  constant  term. 

The  constant  term  cancels  in  all  turn  windings,  as  it  is  equal 
and  opposite  in  the  conductor  and  return  conductor  of  each  turn. 
Direct-current  induction  (continuous,  or  pulsating  current)  thus 
is  possible  only  in  half-turn  windings,  that  is,  windings  in  which 
each  face  conductor  has  a  collector  ring  at  either  end,  so-called 
unipolar  machines  (see  "  Theory  and  Calculation  of  Electrical 

Apparatus")* 

In  every  winding,  which  repeats  at  every  pole  or  180  electrical 
degrees,  as  is  almost  always  the  case,  the  even  harmonics  cancel, 
even  if  they  existed  in  the  face  conductor.  In  any  machine  in 
which  the  flux  distribution  in  successive  poles  is  the  same,  and 
merely  opposite  in  direction,  that  is,  in  which  the  poles  are  symmet- 
rical, no  even  harmonics  are  induced,  as  the  field  flux  distribution 
contains  no  even  harmonics.  Even  harmonics  would,  however, 
exist  in  the  voltage  wave  of  a  machine  designed  as  shown  diagram- 
matically  in  Fig.  52,  as  follows : 

The  south  poles  S  have  about  one-third  the  width  of  the  north 


SHAPING  OF  WAVES 


115 


poles  N,  and  the  armature  winding  is  a  unitooth  50  per  cent,  pitch 
winding,  shown  as  A  in  Fig.  52. 

Assuming  sinusoidal  field  flux  distribution  in  the  air-gaps  under 
the  poles  N  and  S  of  Fig.  52,  curve  I  in  Fig.  53  shows  the  field 
flux  distribution  and  thus  the  voltage  induced  in  a  single-face  con- 
ductor. Curve  II  shows  the  voltage  wave  in  a  50  per  cent,  pitch 
turn  and  therewith  that  of  the  winding  A.  As  seen,  this  contains 
a  pronounced  second  harmonic  in  addition  to  the  fundamental. 
If,  then,  a  second  50  per  cent,  pitch  winding  is  located  on  the  arma- 


FIG.  52. 

ture,  shown  as  B  in  Fig.  52,  by  connecting  B  and  A  in  series  with 
each  other  in  such  direction  that  the  fundamentals  cancel  (that  is, 
in  opposition  for  the  fundamental  wave),  we  get  voltage  wave  III 
of  Fig.  53,  which  contains  only  the  even  harmonics,  that  is,  is  of 
double  frequency.  Connecting  A  and  B  in  series  so  that  the 
fundamentals  add  and  the  second  harmonics  cancel,  gives  the 
wave  IV.  If  the  machine  is  a  three-phase  F-connected  alterna- 
tor, with  curve  IV  as  the  voltage  per  phase,  or  Y  voltage,  the 
delta  or  terminal  voltage,  derived  by  combination  of  two  Y  vol- 
tages under  60°,  then  is  given  by  the  curve  V  of  Fig.  53.  Fig.  54 
shows  the  corresponding  curves  for  the  flux  distribution  of  uni- 
form density  under  the  pole  and  tapering  off  at  the  pole  corners, 
curve  I,  such  as  would  approximately  correspond  to  actual  con- 


116 


ELECTRIC  CIRCUITS 


ditions.  As  seen,  curve  III  as  well  as  V  are  approximately  sine 
waves,  but  the  one  of  twice  the  frequency  of  the  other.  Thus, 
such  a  machine,  by  reversing  connections  between  the  two  wind- 
ings A  and  B,  could  be  made  to  give  two  frequencies,  one  double 
the  other,  or  as  synchronous  motor  could  run  at  two  speeds,  one 
one-half  the  other. 


FIG.  53. 


61.  Distribution  of  the  winding  over  an  arc  of  the  periphery  of 
the  armature  eliminates  or  reduces  the  higher  harmonics,  so  that 
the  terminal  voltage  wave  of  an  alternator  with  distributed  wind- 
ing is  less  distorted,  or  more  nearly  sine-shaped,  than  that  of  a 
single  turn  of  the  same  winding  (or  that  of  a  unitooth  alternator). 
The  voltage  waves  of  successive  turns  are  slightly  out  of  phase 
with  each  other,  and  the  more  rapid  variations  due  to  higher  har- 
monics thus  are  smoothed  out.  In  two  armature  turns  different 


SHAPING  OF  WAVES 


117 


in  position  on  the  armature  circumference  by  5  electrical  degrees 
("electrical  degrees"  means  counting  the  pitch  of  two  poles  as 
360°),  the  fundamental  waves  are  8  degrees  out  of  phase,  the  third 
harmonics  35  degrees,  the  fifth  harmonics  56  degrees,  and  so  on, 
and  their  resultants  thus  get  less  and  less,  and  becomes  zero  for 
that  harmonic  n,  where  nd  =  180°. 


FIG.  54. 


If 


e  =  e\  sn 


£3  sin  3  (*  — 
e7sin7  (*-« 


e6  sin  5  (*— 


(1) 


is  the  voltage  wave  of  a  single  turn,  and  the  armature  winding  of 
m  turns  covers  an  arc  of  o>  electrical  degrees  on  the  armature 
periphery  (per  phase),  the  coefficients  of  the  harmonics  of  the 
resultant  voltage  wave  are 


118 


ELECTRIC  CIRCUITS 


En  =  men  avg.  cos 


nco 


nco 
"2" 


(2) 


or,  since 


avg.  cos 


nco 
~2~ 


2     .     nco 
=  —  sin  — 
nco          2 


nco 

T 


2m  nco 

En  =  --en  sin  — 
nco  2 


(3) 


and 


2m  f         .     co    .  63    .    3co    .     _,  ,  N 

=  —  <  e\  sin  TT  sin  0  +  —  sin  —  sm  3(0  —  a3) 
w    \  -  o          J 

-  a,)  +  . 


(4) 


Thus,  in  a  three-phase  winding  like  that  of  the  three-phase 
synchronous  converter,  in  which  each  phase  covers  an  arc  of  120° 

2r        .    co      TT 
=  -5-,  it  is  ~  =  5,  hence, 


E 


3m\/3 

27T 


sn 


-  ^5  sin  5(0  - 
o 


-I- ^  sin  7(0  -  «7)-  +  ....     1 


(5) 


that  is,  the  third  harmonic  and  all  its  multiples,  the  ninth,  fif- 
teenth, etc.,  cancel,  all  other  harmonics  are  greatly  reduced,  the 
more,  the  higher  their  order. 

In  a  three-phase  Y-connected  winding,  in  which  each  phase 

covers  60°  =  «  of  the  periphery,  as  commonly  used  in  induction 

o 

and  synchronous  machines,  it  is  ^  =  «>  hence, 

4       o 

E  =  —  \  e\  sin  0  +  •=  es  sin  3(0  —  a3)  +  -=  eb  sin  5(0  —  ot6) 

IT       I  O  O 

1  2 

—  j  e-i  sin  7(0  —  a?)  —  g  ^9  sin  9(0  —  «9) 

-  —en  sin  11(0  -  an)  +  j^6is sin  13(0  -  0:13)  +  -  .   .   .  j     (6) 


SHAPING  OF  WAVES  119 

Here  the  third  harmonics  do  not  cancel,  but  are  especially  large. 
Thus  in  a  F-connected  three-phase  machine  of  the  usual  60° 
winding,  the  Y  voltage  may  contain  pronounced  third  harmonics, 
which,  however,  cancel  in  the  delta  voltage. 

Thus  with  the  distributed  armature  winding,  which  is  now  al- 
most exclusively  used,  the  wave-shape  distortion  due  to  the  non- 
sinusoidal  distribution  of  the  field  flux  is  greatly  reduced,  that  is, 
the  higher  harmonics  in  the  voltage  wave  decreased,  the  more  so, 
the  higher  their  order,  and  very  high  harmonics,  such  as  the  seven- 
teenth, thirty-fifth,  etc.,  therefore  do  not  exist  in  such  machines 
to  any  appreciable  extent,  except  where  produced  by  other  causes. 
Such  are  a  pulsation  of  the  magnetic  reluctance  of  the  field  due 
to  the  armature  slots,  or  a  pulsation  of  the  armature  reactance,  as 
discussed  in  Chapter  XXV  of  "Theory  and  Calculation  of  Alter- 
nating-current Phenomena,"  or  a  space  resonance  of  the  armature 
conductors  with  some  of  the  harmonics.  The  latter  may  occur 
if  the  field  flux  distribution  contains  a  harmonic  of  such  order, 
that  the  voltages  induced  by  it  are  in  phase  in  the  successive  arma- 
ture conductors,  and  therefore  add,  that  is,  when  the  spacing  of 
the  armature  conductors  coincides  with  a  harmonic  of  the  field 
flux,  and  the  armature  turn  pitch  and  winding  pitch  are  such  that 
this  harmonic  does  not  cancel. 

Inversely,  if  two  turns  are  displaced  from  each  other  on  the 

armature  periphery  by  -  of  the  pole  pitch,  or  —  ,  and  are  connected 

71  71 

in  series,  then  in  the  resultant  voltage  of  these  two  turns,  the  ntb 
harmonics  are  out  of  phase  by  n  times  -  ,  or  by  TT  =  ISO0,  that  is, 

are  in  opposition  and  so  cancel. 

Thus  in  a  unitooth  F-connected  three-phase  alternator,  while 
each  phase  usually  contains  a  strong  third  harmonic,  the  terminal 
voltage  can  contain  no  third  harmonic  or  its  multiples :  the  two 
phases,  which  are  in  series  between  each  pair  of  terminals,  are 
one-third  pole  pitch,  or  60  electrical  degrees  displaced  on  the 
armature  periphery,  and  their  third  harmonic  voltages  therefore 
3  X  60  =  180°  displaced,  or  opposite,  that  is,  cancel,  and  no  third 
harmonic  can  appear  in  the  terminal  voltage  wave,  or  delta  volt- 
age, but  a  pronounced  third  harmonic  may  exist — and  give 
trouble — in  the  voltage  between  each  terminal  and  the  neutral,  or 
the  F  voltage. 

62.  By  the  use  of  a  fractional-pitch  armature  winding,  higher 
harmonics  can  be  eliminated.  Assume  the  two  sides  of  the  arma- 


120  ELECTRIC  CIRCUITS 

ture  turn,  conductor  and  return  conductor,  are  not  separated 
from  each  other  by  the  full  pitch  of  the  field  pole,  or  180  electrical 
degrees,  but  by  less  (or  more) ;  that  is,  each  armature  turn  or  coil 
covers  not  the  full  pitch  of  the  pole,  but  the  part  p  less  (or  more), 
that  is,  covers  (1  ±  p)  180°.  The  coil  then  is  said  to  be  (1  ±  p) 
fractional  pitch,  or  has  the  pitch  deficiency  p.  The  voltages  in- 
duced in  the  two  sides  of  the  coil  then  are  not  equal  and  in  phase, 
but  are  out  of  phase  by  180  p  for  the  fundamental,  and  by  180 
np  for  the  nth  harmonic.  Thus,  if  np  =  1,  for  this  nih  har- 
monic the  voltages  in  the  two  sides  of  the  coil  are  equal  and  oppo- 
site, thus  cancel,  and  this  harmonic  is  eliminated. 

Therefore,  two-thirds  pitch  winding  eliminates  the  third  har- 
monic, four-fifths  pitch  winding  the  fifth  harmonic,  etc. 

Peripherally  displacing  half  the  field  poles  against  the  other 
half  by  the  fraction  q  of  the  pole  pitch,  or  by  180  q  electrical  de- 
grees, causes  the  voltages  induced  by  the  two  sets  of  field  poles 
to  be  out  of  phase  by  180  nq  for  the  nth  harmonic,  and  thereby 
eliminates  that  harmonic,  for  which  nq  =  1. 

By  these  various  means,  if  so  desired,  a  number  of  harmonics 
can  be  eliminated.  Thus  in  a  F-connected  three-phase  alternator 
with  the  winding  of  each  phase  covering  60  electrical  degrees, 
with  four-fifths  pitch  winding  and  half  the  field  poles  offset  against 
the  other  by  one-seventh  of  the  pole  pitch,  the  third,  fifth,  and 
seventh  harmonic  and  their  multiples  are  eliminated,  that  is,  the 
lowest  harmonic  existing  in  the  terminal  voltage  of  such  a  ma- 
chine is  the  eleventh,  and  the  machine  contains  only  the  eleventh, 
thirteenth,  seventeeth,  ninteenth,  twenty-third,  twenty-ninth, 
thirty-first,  thirty-seventh,  etc.  harmonics.  As  by  the  distrib- 
uted winding  these  harmonics  are  greatly  decreased,  it  follows 
that  the  terminal  voltage  wave  would  be  closely  a  sine,  irrespec- 
tive of  the  field  flux  distribution,  assuming  that  no  slot  harmonics 
exist. 

63.  In  modern  machines,  the  voltage  wave  usually  is  very 
closely  a  sine,  as  the  pronounced  lower  harmonics,  caused  by  the 
field  flux  distribution,  which  gave  the  saw-tooth,  flat-top,  peak 
or  multiple-peak  effects  in  the  former  unitooth  machines,  are 
greatly  reduced  by  the  distributed  winding  and  the  use  of  frac- 
tional pitch.  Individual  high  harmonics,  or  pairs  of  high  harmon- 
ics, are  occasionally  met,  such  as  the  seventeenth  and  ninteenth, 
or  the  thirty-fifth  and  thirty-seventh,  etc.  They  are  due  to  the 
pulsation  of  the  magnetic  field  flux  caused  by  the  pulsation  of  the 


SHAPING  OF  WAVES  121 

field  reluctance  by  the  passage  of  the  armature  slots,  and  occa- 
sionally, under  load,  by  magnetic  saturation  of  the  armature  self- 
inductive  flux,  that  is,  flux  produced  by  the  current  in  an  arma- 
ture slot  and  surrounding  this  slot,  in  cases  where  very  many 
ampere  conductors  are  massed  in  one  slot,  and  the  slot  opening 
bridged  or  nearly  so. 

The  low  harmonics,  third,  fifth,  seventh,  are  relatively  harm- 
less, except  where  very  excessive  and  causing  appreciable  increase 
of  the  maximum  voltage,  or  the  maximum  magnetic  flux  and 
thus  hysteresis  loss.  The  very  high  harmonics  as  a  rule  are  rela- 
tively harmless  in  all  circuits  containing  no  capacity,  since  they 
are  necessarily  fairly  small  and  still  further  suppressed  by  the 
inductance  of  the  circuit.  They  may  become  serious  and  even 
dangerous,  however,  if  capacity  is  present  in  the  circuit,  as  the 
current  taken  by  capacity  is  proportional  to  the  frequency,  and 
even  small  voltage  harmonics,  if  of  very  high  order,  that  is,  high 
frequency,  produce  very  large  currents,  and  these  in  turn  may 
cause  dangerous  voltages  in  inductive  devices  connected  in  series 
into  the  circuit,  such  as  current  transformers,  or  cause  resonance 
effects  in  transformers,  etc.  With  the  increasing  extent  of  very 
high-voltage  transmission,  introducing  capacity  into  the  systems, 
it  thus  becomes  increasingly  important  to  keep  the  very  high 
harmonics  practically  out  of  the  voltage  wave. 

Incidentally  it  follows  herefrom,  that  the  specifications  of  wave 
shape,  that  it  should  be  within  5  per  cent,  of  a  sine  wave,  which  is 
still  occasionally  met,  has  become  irrational :  a  third  harmonic  of 
5  per  cent,  is  practically  negligible,  while  a  thirty-fifth  harmonic  of 
5  per  cent.,  in  the  voltage  wave,  would  hardly  be  permissible.  This 
makes  it  necessary  in  wave-shape  specifications,  to  discriminate 
against  high  harmonics.  One  way  would  be,  to  specify  not  the 
wave  shape  of  the  voltage,  but  that  of  the  current  taken  by  a 
small  condenser  connected  across  the  voltage.  In  the  condenser 
current,  the  voltage  harmonics  are  multiplied  by  their  order. 
That  is,  the  third  harmonic  is  increased  three  times,  the  fifth 
harmonic  five  times,  the  thirty-fifth  harmonic  35  times,  etc. 
However,  this  probably  overemphasizes  the  high  harmonics, 
gives  them  too  much  weight,  and  a  better  way  appears  to  be,  to 
specify  the  current  wave  taken  by  a  small  condenser  having  a 
specified  amount  of  non-inductive  resistance  in  series. 

Thus  for  instance,  if  x  =  1000  ohms  =  capacity  reactance  of 
the  condenser,  at  fundamental  frequency,  r  =  100  ohms  =.  re- 


122  ELECTRIC  CIRCUITS 

sistance  in  series  to  the  condenser,  the  impedance  of  this  circuit, 
for  the  nth  harmonic,  would  be 


.  x  1000  . 

—j  (7) 


or,  absolute,  the  impedance, 


zn  =  1000^  +  0.01  (8) 

and,  the  admittance, 


0.001  n 


+  0.01  n2 
and  therefore,  the  multiplying  factor, 

/-*-•-    /-005"          '•'.        do) 

2/i       Vl  +  0.01  n2 
this  gives,  for 

n  f  n  f 

1  1.0  13  8.0 

3  2.9  15  8.4 

5  4.5  25  9.3 

7  5.8  35  9.6 

9  6.7  45  9.8 

11  7.4  oo  10.0 

Thus,  with  this  proportion  of  resistance  and  capacity,  the  maxi- 
mum intensification  is  tenfold,  for  very  high  harmonics.  By 
using  a  different  value  of  the  resistance,  it  can  be  made  anything 
desired. 

A  convenient  way  of  judging  on  the  joint  effect  of  all  harmonics 
of  a  voltage  wave  is  by  comparing  the  current  taken  by  such  a 
condenser  and  resistance,  with  that  taken  by  the  same  condenser 
and  resistance,  at  a  sine  wave  of  impressed  voltage,  of  the  same 
effective  value. 

Thus,  if  the  voltage  wave 

e  =  600  +  183  +  125  +  97  +  49  +  2n  +  313  +  3023  +  2425 
=  600  {  1  +  0.033  +  0.025  +  0.0157  +  0.00679+  0.0033n 
+  0.005 13  +  0.0523  +  0.0425  1 


SHAPING  OF  WAVES  123 

(where  the  indices  indicate  the  order  of  the  harmonics)  of  effect- 
ive value 

6  =  V6002  +  182  +  122  +  92  +  42  +  22  +  32  +  302  +  242 
=  601.7 

is  impressed  upon  the  condenser  resistance  of  the  admittance,  yn, 
the  current  wave  is 

i  =  0.603  {  1  +  0.0873  +  0.095  +  0.0877  +  0.04459  +  0.0247n 
+  0.0413  +  0.4623  +  0.3725  ) 

=  0.603  X  1.173 
=  0.707 

while  with  a  sine  wave  of  voltage,  of  eQ  =  601.7,  the  current 
would  be 

IQ  =  0.599, 
giving  a  ratio 


or  18  per  cent,  increase  of  current  due  to  wave-shape  distortion  by 
higher  harmonics. 

64.  While  usually  the  sine  wave  is  satisfactory  for  the  purpose 
for  which  alternating  currents  are  used,  there  are  numerous  cases 
where  waves  of  different  shape  are  desirable,  or  even  necessary 
for  accomplishing  the  desired  purpose.  In  other  cases,  by  the 
internal  reactions  of  apparatus,  such  as  magnetic  saturation,  a 
wave-shape  distortion  may  occur  and  requires  consideration  to 
avoid  harmful  results. 

Thus  in  the  regulating  pole  converter  (so-called  "split-pole 
converter")  variations  of  the  direct-current  voltage  are  produced 
at  constant  alternating-current  voltage  input,  by  superposing  a 
third  harmonic  produced  by  the  field  flux  distribution,  as  discussed 
under  "Regulating  Pole  Converter"  in  "  Theory  and  Calcula- 
tion of  Electrical  Apparatus."  In  this  case,  the  third  harmonic 
must  be  restricted  to  the  local  or  converter  circuit  by  proper 
transformer  connections:  either  three-phase  connection  of  the 
converter,  or  Y  or  double-delta  connections  of  the  transformers 
with  a  six-phase  converter. 

The  appearance  of  a  wave-shape  distortion  by  the  third  har- 
monic and  its  multiples,  in  the  neutral  voltage  of  F-connected 
transformers,  and  its  intensifications  by  capacity  in  the  secondary 


124  ELECTRIC  CIRCUITS 

circuit,  and  elimination  by  delta  connection,  has  been  discussed 
in  Chapter  XXV  of  "  Theory  and  Calculation  of  Alternating- 
current  Phenomena." 

In  the  flickering  of  incandescent  lamps,  and  the  steadiness  of 
arc  lamps  at  low  frequencies,  a  difference  exists  between  the  flat- 
top wave  of  current  with  steep  zero,  and  the  peaked  wave  with 
flat  zero,  the  latter  showing  appreciable  flickering  already  at  a 
somewhat  higher  frequency,  as  is  to  be  expected. 

In  general,  where  special  wave  shapes  are  desirable,  they  are 
usually  produced  locally,  and  not  by  the  generator  design,  as 
with  the  increasing  consolidation  of  all  electric  power  supply  in 
large  generating  stations,  it  becomes  less  permissible  to  produce 
a  desired  wave  shape  within  the  generator,  as  this  is  called  upon 
to  supply  power  for  all  purposes,  and  therefore  the  sine  wave  as 
the  standard  is  preferable. 

One  of  the  most  frequent  causes  of  very  pronounced  wave- 
shape distortion,  and  therefore  a  very  convenient  means  of  pro- 
ducing certain  characteristic  deviations  from  sine  shape,  is  mag- 
netic saturation,  and  as  instance  of  a  typical  wave-shape  distor- 
tion, its  causes  and  effects,  this  will  be  more  fully  discussed  in  the 
following. 


CHAPTER  VIII 
SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION 

65.  The  wave  shapes  of  current  or  voltage  produced  by  a  closed 
magnetic  circuit  at  moderate  magnetic  densities,  such  as  are  com- 
monly used  in  transformers  and  other  induction  apparatus,  have 


FIG.  55. 

been  discussed  in  "  Theory  and  Calculation  of  Alternating-cur- 
rent Phenomena. " 

The  characteristic  of  the  wave-shape  distortion  by  magnetic 

125 


126 


ELECTRIC  CIRCUITS 


saturation  in  a  closed  magnetic  circuit  is  the  production  of  a  high 
peak  and  flat  zero,  of  the  current  with  a  sine  wave  of  impressed 
voltage,  of  the  voltage  with  a  sine  wave  of  current  traversing  the 
circuit. 


y 

f 

N 

>N-N 

/, 

/ 

\ 

\ 

\ 

B  =  15.4   =  5.0 
I  =  10.     =  5.0 
lj  =    9.6  =  4.8 
C0=    3.08=1.0 

/ 

'/ 

\ 

\ 

B 

/ 

/ 
l_ 

V 

\ 

/ 

,-"•"* 

'-' 

7 

•--•^ 

-1° 

\ 

\ 
\ 

f~~' 

-—  " 

'/ 

/ 

/ 
/ 

"--.. 

•^~.. 

X 

\ 

„-' 

-"* 

/ 

/ 

/ 

/ 

\ 

\ 

/ 

t 

/ 

/ 

/ 

X 

S 

1 

/ 

/ 

/ 

N 

V 

I 

/ 

X 

% 

x 

,  J 

s 

FIG.  56. 

In  Fig.  55  are  shown  four  magnetic  cycles,  corresponding  re- 
spectively to  beginning  saturation:  B  =  15.4  kilolines  per  cm.2, 
H  =  10;  moderate  saturation:  B  =  17 A,  H  =  20;  high  saturation: 


A 

/< 

/\ 

"N 

\ 

/ 

/ 

\ 

\ 

SB 

B  =  17.4  =  5.0 
I  =  20    =  5.0 
ll=  14.1=3.53 
6ft=  3.48  =  1.0 

/ 

/ 

\ 

I 

\ 

Co 

^ 

V 

\ 

^ 

--- 

£ 

^ 

/  • 

~~"~^~ 

~^~ 

\ 

\ 

^ 

^^ 

^ 

^f 

/ 

/ 

/ 

"^ 

*s^ 

^ 

^ 

\ 
\ 

_^ 

--' 

/ 

/ 

/ 

/ 

"Sr 

^J 

I 

/ 

/ 

/ 

/ 

\ 

\ 

1 

/ 

/ 

/ 

/ 

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1 

/ 
/ 

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•v! 

\/ 

,/ 

FIG.  57. 

B  =  19.0,  H  =  50;  and  very  high  saturation:  B  =  19.7,  H  = 
100.  Figs.  56,  57,  58  and  59  show  the  four  corresponding 
current  waves  7,  at  a  sine  wave  of  impressed  voltage  e0,  and 
therefore  sine  wave  of  magnetic  flux,  B  (neglecting  ir  drop  in 
the  winding,  or  rather,  eQ  is  the  voltage  induced  by  the  alternat- 
ing magnetic  flux  density  B).  In  these  four  figures,  the  maxi- 


SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION  127 

mum  values  of  e0,  B  and  I  are  chosen  of  the  same  scale,  for  wave- 
shape comparison,  though  in  reality,  in  Fig.  59,  very  high  sat- 
uration, the  maximum  of  current,/,  is  ten  times  as  high  as  in  Fig. 
56,  beginning  saturation.  As  seen,  in  Fig.  56  the  current  is  the 
usual  saw-tooth  wave  of  transformer-exciting  current,  but  slightly 
peaked,  while  in  Fig.  59  a  high  peak  exists.  The  numerical 
values  are  given  in  Table  I. 


/-' 

/\ 

"X 

/ 

/ 

\ 

\ 

SB 

B  =  19.0  =  5.0 
I  =  50  -  5.0 
^  =  29.8=2.98 
eo=  3.8=1.0 

/ 

/ 

\ 

\ 

*0 

/ 

/ 

^ 

\ 

I 

\ 

.-" 

_ 

r~~ 

p 

^. 

\ 

\ 

\ 

^ 

---— 

^ 

/ 

/ 

*"-<- 

---.. 

^^ 

—    - 

\ 
-\- 

•^a^r; 

^~ 

•^ 

7 

2 

/ 

/ 

^ 

N 

/ 

/ 

/ 

/ 

\ 

\ 

/ 

/ 

\ 

\ 

f 

/ 

// 

: 

N^ 

\J 

^ 

FIG.  58. 


-    . 

x''* 

/\ 

N 

/ 

^ 

V 

i 

\ 

B 

B  e 

=  19 

.7  = 

=    5. 

1 

/ 

/ 

A 

\ 

1  ' 

-  It 

—    4 

K)   = 

13   « 

•    b.( 
--    2. 

I 

15  * 

e0 

/ 

X* 

^ 

V 

N 

N*l' 

\ 

«o 

=  3.J 

)4  = 

•    1. 

I 

r 

.--—• 

.  —  — 

—    •— 

^ 

^  .^ 

^ 

-^^ 

* 

^ 

X< 

\^ 

-^-- 

---' 

^—— 

•• 

/ 

r 

// 

~-- 

^^.. 

V^ 

--' 

, 

^ 

, 

X^ 

/- 

^ 

•^ 

/ 

\ 

^> 

v^ 

\ 

I 

^ 

^ 

? 

B- 

\ 

\ 

7 

/ 

f 

y 

\ 

\ 

7 

/ 

[x 

' 

s 

S^ 

\/ 

^ 

' 

FIG.  59. 


That  is,  at  beginning  saturation,  the  maximum  value  of  the  saw- 
tooth wave  of  current  differs  little  from  what  it  would  be  with  a 
sine  wave  of  the  same  effective  value,  being  only  4  per  cent, 
higher.  At  moderate  saturation,  however,  the  current  peak  is 
already  42  per  cent,  higher  than  in  a  sine  wave  of  the  same  effective 


128 


ELECTRIC  CIRCUITS 


value,  and  becomes  132  per  cent,  higher  than  in  a  sine  wave,  at 
the  very  high  saturation  of  Fig.  59. 

Inversely,  while  the  maximum  values  of  current  at  the  higher 

TABLE  I 


•  '• 

Begin- 
ning sat- 
uration, 
5  =  15.4 

Moder- 
ate sat- 
uration, 
5  =  17.4 

High 
satura- 
tion, 
5=19.0 

Very  high 
satura- 
tion, 
5  =  19.7 

Sine  wave  of  voltage,  GO,  maximum..  .  . 

3  08 

3  48 

3  80 

3  94 

Maximum  value  of  current   / 

10  00 

20  00 

50  00 

100  00 

Effective  value  of  current,  X  \/2  '  ii  

9.6 

14.1 

29  8 

43  0 

Form  factor  of  current  wave  —  . 

1.04 

1  42 

1  68 

2  32 

ii 
Ratio  of  effective  currents 

1  00 

1  47 

3  11 

4  48 

B  =  15.4  =  5.0 

I  «=  10    -  5.0 

e  -  7.4   -  2.4 

00=  8.08="  1.0 

01-  3.95-  1.282 


\\ 


FIG.  60. 


FIG.  61. 


SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION  129 

saturations  are  two,  five  and  ten  times  the  maximum  current 
value  at  beginning  saturation,  the  effective  values  are  only  1.47, 
3.1  and  4.47  times  higher.  Thus,  with  increasing  magnetic  satura- 
tion, the  effective  value  of  current  rises  much  less  than  the  maxi- 
mum value,  and  when  calculating  the  exciting  current  of  a  satu- 
rated magnetic  circuit,  as  an  overexcited  transformer,  from  the 
magnetic  characteristic  derived  by  direct  current,  under  the  as- 


FIG.  62. 


sumption  of  a  sine  wave,  the  calculated  exciting  current  may  be 
more  than  twice  as  large  as  the  actual  exciting  current. 

66.  Figs.  60  to  63  show,  for  a  sine  wave  of  current,  /,  traversing  a 
closed  magnetic  circuit,  and  the  same  four  magnetic  cycles  given 
in  Fig.  55,  the  waves  of  magnetic  flux  density,  B,  of  induced  vol- 
tage, e,  the  sine  wave  of  voltage,  60,  which  would  be  induced  if  the 

9 


130 


ELECTRIC  CIRCUITS 


magnetic  density,  5,  were  a  sine  wave  of  the  same  maximum  value, 
and  Fig.  63  also  shows  the  equivalent  sine  wave,  ei,  of  the 
(distorted)  induced  voltage  wave,  e. 

As  seen,  already  at  beginning  saturation,  Fig.  60,  the  voltage 
peak  is  more  than  twice  as  high  as  it  would  be  with  a  sine  wave, 


B  =  19.7  <=  5.0 
I  =100.  =  5.0 

=  73.  =  18.5 

-  3.94-  1.0 

-  13.8  -  3. 


FIG.  63. 

and  rises  at  higher  saturations  to  enormous  values:  18.5  times  the 
sine  wave  value  in  Fig.  63. 

The  magnetic  flux  wave,  B,  becomes  more  and  more  flat-topped 
with  increasing  saturation,  and  finally  practically  rectangular,  in 
Fig.  63. 

The  curves  60  to  63  are  drawn  with  the  same  maximum  values 


SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION  131 


of  current,  7,  flux  density,  B,  and  sine  wave  voltage,  eQ,  for  better 
comparison  of  their  wave  shapes. 
The  numerical  values  are : 

TABLE  II 


Begin- 
ning sat- 
uration, 
B  =  15A 

Moder- 
ate sat- 
uration, 
5  =  17.4 

High 
satura- 
tion, 
5  =  19.0 

Very  high 
satura- 
tion, 
5  =  19.7 

Sine  wave  of  current,  /,  maximum 

10   0 

20  0 

50   0 

100  0 

Flat-top   wave   of  magnetic  density,  B, 
maximum 

15.4 

17.4 

19.0 

19.7 

Peaked  voltage  wave  e,  maximum            .  . 

7.4 

18  8 

35  5 

73  0 

Ratio  

1.00 

2.56 

4.80 

9.88 

Sine  wave  of  voltage,  Co,  maximum,  for 
same  maximum  flux                        

3  08 

3  48 

3  80 

3  94 

Ratio                    

1.00 

1.13 

1.23 

1.28 

Form  factor  of  voltage  wave,  —  

2.40 

5.40 

9.35 

18.50 

Co 
Equivalent  sine  wave  of  voltage,  e\,  maxi- 
mum      

3.95 

6.33 

9.58 

13.80 

Ratio                                     

1.00 

1.60 

2.42 

3.50 

d  , 
—  (maxima)                                  

1  282 

1.864 

2.520 

3.500 

—  (maxima) 

1  87 

2  97 

3  70 

5  28 

e\ 

As  seen,  the  wave-shape  distortion  due  to  magnetic  saturation 
is  very  much  greater  with  a  sine  wave  of  current  traversing  the 
closed  magnetic  circuit,  than  it  is  with  a  sine  wave  of  voltage  im- 
pressed upon  it. 

With  increasing  magnetic  saturation,  with  a  sine  wave  of  cur- 
rent, the  effective  value  of  induced  voltage  increases  much  more 
rapidly  than  the  magnetic  flux  increases,  and  the  maximum  value 
of  voltage  increases  still  much  more  rapidly  than  the  effective 
value:  an  increase  of  flux  density,  B,  by  28  per  cent.,  from  begin- 
ning to  very  high  saturation,  gives  an  increase  of  the  effective 
value  of  induced  voltage  (as  measured  by  voltmeter)  by  250  per 
cent.,  or  3.5  times,  and  an  increase  of  the  peak  value  of  voltage 
(which  makes  itself  felt  by  disruption  of  insulation,  by  danger  to 
life,  etc.)  by  888  per  cent.,  or  nearly  ten  times. 

At  very  high  saturation,  the  voltage  wave  practically  becomes 
one  single  extremely  high  and  very  narrow  voltage  peak,  which 
occurs  at  the  reversal  of  current. 


132  ELECTRIC  CIRCUITS 

At  the  very  high  saturation,  Fig.  63,  the  effective  value,  e\,  of 
the  voltage  is  3.5  times  as  high  as  it  would  be  with  a  sine  wave  of 
magnetic  flux;  the  maximum  value,  e,  is  more  than  five  times  as 
high  as  it  would  be  with  a  sine  wave  of  the  same  effective  value, 
Ci,  that  is,  more  than  five  times  as  high,  as  would  be  expected 
from  the  voltmeter  reading,  and  it  is  18.5  times  as  high  as  it 
would  be  with  a  sine  wave  of  magnetic  flux. 

Thus,  an  oversaturated  closed  magnetic  circuit  reactance, 
which  consumes  eQ  =  50  volts  with  a  sine  wave  of  voltage,  eQ,  and 
thus  of  magnetic  density,  B,  would,  at  the  same  maximum  mag- 
netic density,  that  is,  the  same  saturation,  with  a  sine  wave  of 
current — as  would  be  the  case  if  the  reactance  is  connected  in  ser- 
ies in  a  constant-current  circuit — give  an  effective  value  of  ter- 
minal voltage  of  ei  =  3.5  X  50  =  175  volts,  and  a  maximum  peak 
voltage  of  e  =  18.8  X  50  X  \/2  =  1330  volts. 

Thus,  while  supposed  to  be  a  low-voltage  reactance,  e0  =  50 
volts,  and  even  the  voltmeter  shows  a  voltage  of  only  e\  —  175, 
which,  while  much  higher,  is  still  within  the  limit  that  does  not 
endanger  life,  the  actual  peak  voltage  e  =  1330  is  beyond  the 
danger  limit. 

Thus,  magnetic  saturation  may  in  supposedly  low-voltage  cir- 
cuits produce  dangerously  high-voltage  peaks. 

A  transformer,  at  open  secondary  circuit,  is  a  closed  magnetic 
circuit  reactance,  and  in  a  transformer  connected  in  series  into  a 
circuit — such  as  a  current  transformer,  etc. — at  open  secondary 
circuit  unexpectedly  high  voltages  may  appear  by  magnetic 
saturation. 

67.  From  the  preceding,  it  follows  that  the  relation  of  alternat- 
ing current  to  alternating  voltage,  that  is,  the  reactance  of  a  closed 
magnetic  circuit,  within  the  range  of  magnetic  saturation,  is  not 
constant,  but  varies  not  only  with  the  magnetic  density,  B,  but  for 
the  same  magnetic  density  B,  the  reactance  may  have  very  differ- 
ent values,  depending  on  the  conditions  of  the  circuit:  whether 
constant  potential,  that  is,  a  sine  wave  of  voltage  impressed  upon 
the  reactance;  or  constant  current,  that  is,  a  sine  wave  of  current 
traversing  the  circuit;  or  any  intermediate  condition,  such  as 
brought  about  by  the  insertion  of  various  amounts  of  resistance, 
or  of  reactance  or  capacity,  in  series  to  the  closed  magnetic  cir- 
cuit reactance. 

The  numerical  values  in  Table  III  illustrate  this. 

/  gives  the  magnetic  field  intensity,  and  thus  the  direct  current, 


SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION  133 

which  produces  the  magnetic  density,  B — that  is,  the  B-H 
curve  of  the  magnetic  material.  An  alternating  current  of  maxi- 
mum value,  /,  thus  gives  an  alternating  magnetic  flux  of  maxi- 
mum flux  density  B.  If  /  and  B,  were  both  sine  waves,  that  is,  if 


20   30   40 


in 


IV 


II 


20 


18 


JL7- 
_16 
-15. 
_U 


70   80   90  100  110  120  130  140  150 


FIG.  64. 


during  the  cycle  current  and  magnetic  flux  were  proportional  to 
each  other,  as  in  an  unsaturated  open  magnetic  circuit,  e0,  as  given 
in  the  third  column,  would  be  the  maximum  value  of  the  induced 

voltage,  and  XQ  =  y  the  reactance.     This  reactance  varies  with 


134 


ELECTRIC  CIRCUITS 


the  density,  and  greatly  decreases  with  increasing  magnetic  satu- 
ration, as  well  known. 

However,  if  e0  and  thus  B  are  sine  waves,  I  can  not  be  a  sine 
wave,  but  is  distorted  as  shown  in  Figs.  56  to  59,  and  the  effective 
value  of  the  current,  that  is,  the  current  as  it  would  be  read  by  an 
alternating  ammeter,  multiplied  by  \/2  (that  is,  the  maximum 
value  of  the  equivalent  sine  waves  of  exciting  current)  is  given  as 

ii.     The  reactance  is  then  found  as  xp  =  — •     This  is  the  reactance 


FIG.  65. 

of  the  closed  magnetic  circuit  on  constant  potential,  that  is,  on  a 
sine  wave  of  impressed  voltage,  and,  as  seen,  is  larger  than  x0. 

If,  however,  the  current,  /,  which  traverses  the  reactance,  is  a 
sine  wave,  then  the  flux  density,  B,  and  the  induced  voltage  are  not 
sines,  but  are  distorted  as  in  Figs.  60  to  63,  and  the  effective  value 
of  the  induced  voltage  (that  is,  the  voltage  as  read  by  alternating 
voltmeter) ,  multiplied  by  \/2  (that  is ;  the  maximum  of  the  equiva- 
lent sine  wave  of  voltage)  is  given  as  e\  in  Table  III,  and  the  true 
maximum  value  of  the  induced  voltage  wave  is  e. 


SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION  135 


The  reactance,  as  derived  by  voltmeter  and  ammeter  readings 
under  these  conditions,  that  is,  on  a  constant-current  circuit,  or 
with  a  sine  wave  of  current  traversing  the  magnetic  circuit,  is 


e\ 


Xc  =  —,  thus  larger  than  the  constant-potential  reactance,  xp. 
Much  larger  still  is  the  reactance  derived  from  the  actual  maxi- 

/> 

mum  values  of  voltage  and  current:  xm  =     - 


\ 

v 

\ 

Xm 

—           _ 

• 

—          — 

—         — 

—        __ 

"          — 

\ 

\ 

\ 

\ 

\ 

\ 

V 

\ 

XC 

\ 

Xp 

\ 

X0 

\N 

\ 

\ 

\ 

\ 
\ 

\ 

s 

s 

\ 

\ 

\ 

\ 

\ 

\ 

s 

x> 

B  — 

—  > 

s 

\ 

1 

! 

1 

0      1 

l     1 

2      1 

•i     1 

1      1 

5     1 

5      1 

7      1 

3      1 

sj 

9      2 

0 

FIG.  66. 

It  is  interesting  to  note  that  xmj  the  peak  reactance,  is  approxi- 
mately constant,  that  is,  does  not  decrease  with  increasing  mag- 
netic saturation.  (The  higher  value  at  beginning  saturation, 
for  I  =  20,  may  possibly  be  due  to  an  inaccuracy  in  the  hysteresis 
cycle  of  Fig.  55,  a  too  great  steepness  near  the  zero  value,  rather 
than  being  actual.) 

It  is  interesting  to  realize,  that  when  measuring  the  reactance 
of  a  closed  magnetic  circuit  reactor  by  voltmeter  and  ammeter 
readings,  it  is  not  permissible  to  vary  the  voltage  by  series  resist- 
ance, as  this  would  give  values  indefinite  between  xp  and  xc,  de- 


136 


ELECTRIC  CIRCUITS 


pending  on  the  relative  amount  of  resistance.  To  get  xp,  the 
generated  supply  voltage  of  a  constant-potential  source  must  be 
varied;  to  get  xc,  the  current  in  a  constant-current  circuit  must 
be  varied.  As  seen,  the  differences  may  amount  to  several  hun- 
dred per  cent. 

As  graphical  illustration,  Fig.  64  shows: 

As  curve  I  the  magnetic  characteristic,  as  derived  with  direct 
current. 

Curve  II  the  volt-ampere  characteristic  of  the  closed  circuit 
reactance,  /,  eQ)  as  it  would  be  if  /  and  B,  that  is,  e0,  both  were  sine 
waves. 

Curve  III  the  volt-ampere  characteristic  on  constant-potential 
alternating  supply,  i\,  e0. 

Curve  IV  the  volt-ampere  characteristic  on  constant-current 
alternating  supply,  as  derived  by  voltmeter  and  ammeter,  7, 
Ci,  and  as 

Curve  V  the  volt-ampere  characteristic  on  constant-current 
alternating  supply,  as  given  by  the  peak  values  of  /  and  e. 

Fig.  65  gives  the  same  curves  in  reduced  scale,  so  as  to  show  V 
completely. 

Fig.  66  then  shows,  with  B  as  abscissae,  the  values  of  the  react- 
ances XQ}  xp}  xc,  and  xm. 

TABLE  III 


I 

B 

eg 

eo 

*0=7 

ii 

eo 
Xp   =  iT 

ei 

e\ 

Xc    =   -J 

e 

e 

Xm=j 

P 

Po 

2  0 

7  30 

0  7300 

1  00 

3  0 

10  00 

0  6670 

1  09 

4  0 

11   50 

0  5750 

1.27 

5  0 

12  50 

0  5000 

1.46 

7  5 

14  30 

0  3810 

1.92 

10.0 
15  0 

15.40 
16  70 

3.08 

0.3080 
0  2230 

9.0 

0.342 

3.95 

0.395 

7.4 

0.74 

2.37 
3.27 

2.40 

20.0 
30  0 

17.40 
18  30 

3.48 

0.1740 
0   1220 

14.1 

0.247 

6.33 

0.316 

18.8 

0.94 

4.20 
6  00 

5.40 

40  0 

18  70 

0  0930 

7.85 

50.0 
75  0 

19.00 
19  35 

3.80 

0  .  0760 
0  0520 

29.8 

0.127 

9.58 

0.912 

35.5 

0.71 

9.60 
14.10 

9.35 

100.0 
125  0 

19.70 
19  85 

3.94 

0.0394 
0  0320 

43.0 

0.092 

13.80 

0.138 

73.0 

0.73 

18.50 
22.80 

18.5 

150  0 

19  95 

0  0270 

27.00 

68.  Another  way  of  looking  at  the  phenomenon  is  this:  while 
with  increasing  current  traversing  a  closed  magnetic  circuit,  the 
magnetic  flux  density  is  limited  by  saturation,  the  induced  voltage 


SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION  137 

peak  is  not  limited  by  saturation,  as  it  occurs  at  the  current 'rever- 
sal, but  it  is  proportional  to  the  rate  of  change  of  the  magnetic 
flux  density  at  the  current  reversal,  and  thus  approximately  pro- 
portional to  the  current. 

Thus,  approximately,  within  the  range  of  magnetic  saturation, 
with  increasing  current  traversing  the  closed  magnetic  circuit 
(like  that  of  a  series  transformer) : 

The  magnetic  flux  density,  and  therefore  the  mean  value  of  in- 
duced voltage  remains  constant; 

The  peak  value  of  induced  voltage  increases  proportional  to  the 
current,  and  therefore; 

The  effective  value  of  induced  voltage  increases  proportional 
to  the  square  root  of  the  current. 

Thus,  if  the  exciting  current  of  a  series  transformer  is  5  per  cent, 
of  full-load  current,  and  the  secondary  circuit  is  opened,  while  the 
primary  current  remains  the  same,  the  effective  voltage  consumed 
by  the  transformer  increases  approximately  \/20  =  4.47  times, 
and  the  maximum  voltage  peak  20  times  above  the  full-load 
voltage  of  the  transformer. 

As  the  shape  of  the  magnetic  flux  density  and  voltage  waves  are 
determined  by  the  current  and  flux  relation  of  the  hysteresis  cy- 
cles, and  the  latter  are  entirely  empirical  and  can  not  be  expressed 
mathematically,  therefore  it  is  not  possible  to  derive  an  exact 
mathematical  equation  for  these  distorted  and  peaked  voltage 
waves  from  their  origin.  Nevertheless,  especially  at  higher  satu- 
ration, where  the  voltage  peaks  are  more  pronounced,  the  equa- 
tion of  the  voltage  wave  can  be  derived  and  represented  by  a 
Fourier  series  with  a  fair  degree  of  accuracy.  By  thus  deriving 
the  Fourier  series  which  represents  the  peaked  voltage  waves,  the 
harmonics  which  make  up  the  wave,  and  their  approximate  val- 
ues can  be  determined  and  therefrom  their  probable  effect  on  the 
system,  as  resonance  phenomena,  etc.,  estimated. 

The  characteristic  of  the  voltage-wave  distortion  due  to  mag- 
netic saturation  in  a  closed  magnetic  circuit  traversed  by  a  sine 
wave  of  current  is,  that  the  entire  voltage  wave  practically  con- 
tracts into  a  single  high  peak,  at,  or  rather  shortly  after,  the  mo- 
ment of  current  reversal,  as  shown  in  Figs.  63,  62,  etc. 

With  the  same  maximum  value  of  magnetic  density,  B,  and  thus 
of  flux,  3>,  the  area  of  the  induced  voltage  wave,  and  thus  the  mean 
value  of  the  voltage,  is  the  same,  whatever  may  be  the  wave  of 
magnetism  and  thus  of  voltage,  since  <£  =  J  e  dt,  and  the  area  of 


138  ELECTRIC  CIRCUITS 

the  peaked  voltage  wave  of  the  saturated  magnetic  circuit,  e,  thus 
is  the  same  as  that  of  a  sine  wave  of  voltage,  e0.  Neglecting  then 
the  small  values  of  voltage,  e,  outside  of  the  voltage  peak,  if  this 
voltage  peak  of  e  is  p  times  the  maximum  value  of  the  sine  wave, 

eQ,  its  width  is  -  of  that  of  the  sine  wave,  and  if  the  sine  wave  of 
voltage,  e0,  is  represented  by  the  equation 

€Q  COS  0  (11) 

the  peak  of  the  distorted  voltage  wave  is  represented,  in  first  ap- 
proximation, by  assuming  it  as  of  sinusoidal  shape,  by 

pe0  cos  p<f>  (12) 

That  is,  the  distorted  voltage  wave,  e,  can  be  considered  as 
represented  by  peQ  cos  p<f>  within  the  angle 

..  -£<*<£    •    ••':;     <13) 

and  by  zero  outside  of  this  range. 

The  value  of  p  follows,  approximately,  from  the  consideration 
that  the  peak  reactance,  xm,  is  independent  of  the  saturation,  or 
constant,  since  it  depends  on  the  rate  of  change  of  magnetism 
with  current  near  the  zero  value,  where  there  is  no  saturation,  and 

the  ratio  -TF  thus  (approximately)  constant. 

Or,  in  other  words,  if  below  saturation,  in  the  range  where  the 
magnetic  permeability  is  a  maximum,  the  current,  i,  produces  the 
magnetic  flux,  3>,  and  thereby  induces  the  voltage,  e',  the  reactance 
is 

*'  =  -{  (14) 

This  is  the  maximum  reactance,  below  saturation,  of  the  mag- 
netic circuit,  and  can  be  calculated  from  the  dimensions  and  the 
magnetic  characteristic,  in  the  usual  manner,  by  assuming  sine 
waves  of  i  and  B. 

The  peak  reactance,  xm,  of  the  saturated  magnetic  circuit  is  ap- 
proximately equal  to  x',  and  thus  can  be  calculated  with  reason- 
able approximation,  from  the  dimensions  of  the  magnetic  circuit 
and  the  magnetic  characteristic. 

If  now,  in  the  range  of  magnetic  saturation,  a  sine  wave  of  cur- 


SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION  139 

rent,  of  maximum  value  /,  traverses  the  closed  magnetic  circuit, 
the  peak  value  of  the  (distorted)  induced  voltage  is 

e  =  xml  (15) 

where 

Xm    =    X'    =    ~.  (16) 

is  the  maximum  reactance  of  the  magnetic  circuit  below  satura- 
tion, derived  by  the  assumption  of  sine  waves,  e'  and  i. 

If  B  is  the  maximum  value  of  the  magnetic  density  produced  by 
the  sine  wave  of  current  of  maximum  value,  7,  and,  e0,  the  maxi- 
mum value  of  the  sine  wave  of  voltage  induced  by  a  sinusoidal 
variation  of  the  magnetic  density,  B,  the  "form  factor"  of  the 
peaked  voltage  wave  of  the  saturated  magnetic  circuit  is 

^  =  XrJ 

e0        e0 

thus  determined,  approximately. 

As  illustrations  are  given,  in  the  second  last  column  of  Table 
III,  the  form  factors,  p,  calculated  in  this  manner,  and  in  the  last 
column  are  given  the  actual  form  factors,  p0,  derived  from  the 
curves  60  to  63.  As  seen,  the  agreement  is  well  within  the  un- 
certainty of  observation  of  the  shape  of  the  hysteresis  cycles, 
except  perhaps  at  7  =  20,  and  there  probably  the  calculated 
value  is  more  nearly  correct. 

69.  The  peaked  voltage  wave  induced  by  the  saturated  closed 
magnetic  circuit  can,  by  assuming  it  as  symmetrical  and  counting 
the  time  from  the  center  of  the  peak,  be  represented  by  the 
Fourier  series. 


e  =  di  cos  </>  +  a3  cos  3  <f>  +  a5  cos  5  <£  +  a7  cos  7  0 

=  2  an  cos  n<£ 
where 

e  cos  n<f>d<f> 


(18) 


=  2  avg(e  cos  n<t>)%  (20) 

The  slight  asymmetry  of  the  peak  would  introduce  some  sine 
terms,  which  might  be  evaluated,  but  are  of  such  small  values  as 
to  be  negligible. 

(a)  For  the  lower  harmonics,  where  n  is  small  compared  to  p, 


140  ELECTRIC  CIRCUITS 

cos  n<f>  is  practically  constant  and  =  1  during  the  short  voltage 
peak  e  =  peQ  cos  p<f>}  and  it  is,  therefore, 

an  =  2avg(e)o 

IT 

=  2  avg(pe0  cos 

=  -  avg(pe0  cos 

4 

=  2  60  avg  cos  =  -  e0. 

(b)  For  the  harmonic,  where  n  =  p,  it  is 

•K 

ap  =  2  avg(pe0  cos2 

2 

=  -  avg(pe0  cos2 

=  2  eQ  avg  cos2  =  e0. 

(c)  For  still  higher  harmonics  than  n  =p,  cos   n<f>   assumes 
negative  values  within  the  range  of  the  voltage  peak,  and  an 
thereby  rapidly  decreases,  finally  becomes  zero  and  then  negative, 
at  n  =  3  p}  positive  again  at  n  =  5  p,  etc.,  but  is  practically 
negligible. 

Thus,  the  coefficients  of  the  Fourier  series  decrease  gradually, 

with  increasing  order,  n, 

4 

from  -  60  as   maximum,   to  e0   for  n  =  p,   and  then   with   in- 
creasing rapidity  fall  off  to  negligible  values. 

Their  exact  values  can  easily  be  derived  by  substituting  (12) 
into  (19), 

TT 

4  |    P  f21  ^ 

an  —  -        peo  cos  p<t>  cos  n<f>d<t) 

7T 

here  the  integration  is  extended  to  ^—  only,  as  beyond  this, .the 

voltage,  6,  is  not  given  by  equation  (12)  any  more,  but  is  zero. 
(21)  integrates  by 

•K 

2  peQ  /sin(p  +  n)</>       sin(p  —  n)<J>  /2p 
TT/        p  +  n  p  —  n      /Q 


p  +  n  p  —  n 


SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION  141 

but  since        sin  ^U  H — )  =  sin  ~  (l j,  it  it 

.    TT/,        n\ 
4  60  sm -(!--) 


and 


s 


e  = 


cos 


(22) 


(23) 


as  the  equations  of  the  voltage  wave  distorted  by  magnetic 
saturation. 

70.  These  coefficients,  an,  are  very  easily  calculated,  and  as  in- 
stances are  given  in  Table  IV,  the  coefficients  of  the  distorted 
voltage  wave  of  Fig.  62,  which  has  the  form  factor  p  =  9.35. 


TABLE  IV 


p  =  9.35 


an 


n  =  1 
—  =  1.270 

3 
1.242 

5 
1.188 

7 
1.114 

9 
1.018 

11 
0.906 

13 

0.786 

15 
0.658 

17 
0.528 

19 
0.406 

n  =  21 

23 

25 

27 

29 

31 

33 

—  =  0.292 

0.189 

0.101 

0.031 

-0.023 

-0.060 

-0.082 

As  seen,  after  n  =  9,  the  values  of  an  rapidly  decrease,  and  be- 
come negative,  though  of  negligible  value,  after  n  =  27. 

In  Fig.  67  the  successive  values  of  —  are  shown  as  curve. 

£o 

In  reality,  the  peaked  voltage  wave  of  magnetic  saturation,  as 
shown  in  Figs.  61  to  63,  is  not  half  a  sine  wave,  but  is  rounded  off 
at  the  ends,  toward  the  zero  values.  Physically,  the  meaning  of 
the  successive  harmonics  is,  that  they  raise  the  peak  and  cut  off 
the  values  outside  of  the  peak.  It  is  the  high  harmonics,  which 
sharpen  the  edge  of  the  peak,  and  the  rounded  edge  of  the  peak  in 
the  actual  wave  thus  means  that  the  highest  harmonics,  which 
give  very  small  or  negative  values  of  an;  are  lower  than  given  by 
equations  (23),  or  rather  are  absent. 


142 


ELECTRIC  CIRCUITS 


Thus,  by  omitting  the  highest  harmonics,  the  wave  is  rounded 
off  and  brought  nearer  to  its  actual  shape.  Thus,  instead  of  fol- 
lowing the  curve,  an,  as  calculated  and  given  in  Fig.  67,  we  cut  it 
off  before  the  zero  value  of  an,  about  at  n  =  23,  and  follow  the 

curve  line,  a'n,  which  is  drawn  so  that  S—  =  9.35,  that  is,  that 
the  voltage  peak  has  the  actual  value. 


-.6 

s^ 

1.2 

N 

1.1 

.B 

\ 

an 

4     SinJ?[(l  — 

P  \ 
n  t 

1  0 

\ 

€Q          7T 

l-Po 

n- 

.9 

..4 

\ 

.8 

an 

e  o 

\ 

.7 

-3 

\ 

V 

.6 

\ 

.5 

? 

\ 

V 

.3 

,1 

IE 

\ 

2 

\ 

\ 

0 

\ 

\ 

x 

0 

1 

J        ! 

i 

1 

1      l 

3      1 

5      1 

7       1 

9      2 

1  _J 

^ 

5      2 

7     ^ 

9^< 

1    33 

. 

.1 

.2 

1      J 

j 

j 

£ 

1 

L      1 

3      1 

5      1 

7      1 

3      2 

L      2 

3      2 

5      2 

r    2 

9      3 

33 

FIG.  67. 
The  equation  of  the  peaked  voltage  in  Fig.  62  then  becomes 

e  =  e0  {  1.270  cos  <j>  +  1.242  cos  3<£  +  1.188  cos  50  +  1.114  cos  70 
+  1.018  cos  90  +  0.906  cos  110  +  0.786  cos  130  +  0.658  cos  150 
+  0.529  cos  170  +  0.400  cos  190  +  0.240  cos  210). 

Or,  in  symbolic  writing, 


e0{1.270i  + 
+  0.786ia 


l-2423  +  1.188B  +  1. 
+  0.658i5  +  0.529i7 


1.0189  +  0.906n 
0.40019  +  0.24021 


SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION  143 


1.270  e0  {li 
+  0.617i3 


0.9783 
0.517i5 


0.9535  +  0.8777  +  0.8009  +  0.713n 
0.416i7  +  0.31519  +  0.1892i). 


It  is  of  interest  to  note  how  extended  a  series  of  powerful  har- 
monics is  produced.  It  is  easily  seen  that  in  the  presence  of  ca- 
pacity, these  large  and  very  high  harmonics  may  be  of  consider- 
able danger.  In  any  reactance,  which  is  intended  for  use  in  series 
to  a  high-  voltage  circuit,  the  use  of  a  closed  magnetic  circuit  thus 
constitutes  a  possible  menace  from  excessive  voltage  peaks  if 
saturation  occurs. 


FIG.  68. 

71.  Such  high- voltage  peaks  by  magnetic  saturation  in  a  closed 
magnetic  circuit  traversed  by  a  sine  wave  of  current  can  occur 
only  if  the  available  supply  voltage  is  sufficiently  high.  If  the 
total  supply  voltage  of  the  circuit  is  less  than  the  voltage  peak  pro- 
duced by  magnetic  saturation,  obviously  this  voltage  peak  must 
be  reduced  to  a  value  below  the  voltage  available  in  the  supply 
circuit,  and  in  this  case  simply  the  current  wave  can  not  remain 
a  sine,  but  is  flattened  at  the  zero  values,  and  with  it  the  wave  of 
magnetic  density. 

Thus,  if  in  Fig.  62  the  maximum  supply  voltage  is  E  =  19.0, 
the  maximum  peak  voltage  can  not  rise  to  e  =  35.5,  but  stops  at 


144  ELECTRIC  CIRCUITS 


and  when  this  value  is  reached,  the  rate  of  change  of  flux 
density,  B,  and  thus  of  current,  /,  decreases,  as  shown  in  Fig.  68, 
in  drawn  lines.  In  dotted  lines  are  added  the  curves  correspond- 
ing to  unlimited  supply  voltage.  The  voltage  peak  is  thereby 
reduced,  correspondingly  broadened,  and  retarded,  and  the  cur- 
rent is  flattened  at  and  after  its  zero  value,  the  more,  the  lower 
the  maximum  supply  voltage. 

The  reactance  is  reduced  hereby  also,  from  xc  =  0.192,  in  Fig. 
62,  to  xc  =  0.140. 

In  other  words,  if  p  is  the  form  factor  of  the  distorted  voltage 
wave,  which  would,  with  unlimited  supply  voltage,  be  induced  by 
the  saturated  magnetic  circuit  of  maximum  density,  B,  and  eQ  is 
the  maximum  value  of  the  sine  wave  of  voltage,  which  a  sinu- 
soidal flux  of  maximum  density,  B}  would  induce,  the  distorted 
voltage  peak  is 

e  =  pe0  (24) 

and  the  maximum  value  of  the  equivalent  sine  wave  of  the  dis- 
torted voltage,  or  the  effective  voltage  read  by  voltmeter,  is 


(25) 
If  now  the  maximum  voltage  peak  is  cut  down  to  E,  by  the 

p 

limitation  of  the  supply  voltage,  and  -^  =  q,  the  form  factor  be- 
comes 

p'  =  ^  =  2,  (26) 

e0       q 

and  the  effective  value  of  the  distorted  voltage,  times  \/2  ,  that  is, 
the  maximum  of  the  equivalent  sine  wave,  is 


_ 

=  Ve0E, 

thus  varies  with  the  supply  voltage,  E. 
The  reactance  then  is 


Thus,  for  e  =  35.5,  E  =  19.0,  it  is 

q  =  1-87, 


SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION  145 
and  as  e0  =  3.80;  p  =  9.35,  it  is 


2          9'58        70 

'  '  =  —       L37       7'°' 


*.  =  -%  =  2^2  _  L4a 


These  values,  however,  are  only  fair  approximations,  as  they 
are  based  on  the  assumption 
of    sinusoidal    shape    of    the 
peaks. 

72.  In  the  preceding,  the 
assumption  has  been  made, 
that  the  magnetic  flux  passes 
entirely  within  the  closed 
magnetic  circuit,  that  is,  that 
there  is  no  magnetic  leakage 
flux,  or  flux  which  closes 
through  non-magnetic  space 
outside  of  the  iron  conduit. 

If  there  is  a  magnetic  leak- 
age flux — and  there  must 
always  be  some — it  somewhat 
reduces  the  voltage  peak,  the 
more,  the  greater  is  the  pro- 
portion of  the  leakage  flux  to 
the  main  flux.  The  leakage 
flux,  in  open  magnetic  circuit, 
is  practically  proportional  to 
the  current,  and  that  part  of 
the  voltage,  which  is  induced 
by  the  leakage  flux,  therefore, 
is  a  sine  wave,  with  a  sine 
wave  of  current,  hence  does  not  FIG.  69. 

contribute  to  the  voltage  peak. 

Such  high  magnetic  saturation  peaks  occur  only  in  a  closed 
magnetic  circuit.  If  the  magnetic  circuit  is  not  closed,  but  con- 
tains an  air-gap,  even  a  very  small  one,  the  voltage  peak,  with  a 
sine  wave  of  current,  is  very  greatly  reduced,  since  in  the  air-gap 
magnetic  flux  and  magnetizing  current  are  proportional. 
10 


146 


ELECTRIC  CIRCUITS 


Thus,  below  saturation  and  even  at  beginning  saturation,  an 
air-gap  in  the  magnetic  circuit,  of  one-hundredth  of  its  length, 
makes  the  voltage  wave  practically  a  sine  wave,  with  a  sine  wave 
of  current,  as  discussed  in  "Theory  and  Calculation  of  Alternating- 
current  Phenomena." 


7 


\ 


FIG.  70. 


The  enormous  reduction  of  the  voltage  peak  by  an  air-gap  of 
1  per  cent,  of  the  length  of  the  magnetic  circuit  is  shown  in  Figs. 
69  and  70. 

In  Fig.  69,  with  the  magnetic  flux  density,  B,  as  abscissae,  the 


SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION  147 

m.m.f.  of  the  iron  part  of  the  magnetic  circuit  is  shown  as  curve  I. 
This  would  be  the  magnetizing  current  if  the  magnetic  circuit 
were  closed.  Curve  II  show  the  m.m.f.  consumed  in  an  air-gap 
of  1  per  cent,  of  the  length  of  the  magnetic  circuit  of  curve  I, 
and  curve  III,  therefore,  shows  the  total  m.m.f.  of  the  magnetizing 
current  of  the  magnetic  circuit  with  1  per  cent,  air-gap. 

Choosing  as  instance  the  very  high  saturation  B  =  19.7,  the 
same  as  illustrated  in  Fig.  63,  and  neglecting  the  hysteresis — 
which  is  permissible,  as  the  hysteresis  does  not  much  contribute 
to  the  wave-shape  distortion — the  corresponding  voltage  waves 
are  plotted  in  Fig.  70,  in  the  same  scale  as  Figs.  56  to  63:  for  a 
sine  wave  of  current,  curves  Fig.  69  give  the  corresponding  values 
of  magnetic  flux,  and  from  the  magnetic  flux  wave  is  derived,  as 

-r-,  the  voltage  wave.  The  waves  of  magnetism  are  not  plotted. 
ckp 

CQ  is  the  sine  wave  of  voltage,  which  would  be  induced  by  a  sinu- 
soidal variation  of  magnetic  flux;  e  is  the  peaked  voltage  wave 
induced  in  a  closed  magnetic  circuit  of  the  same  maximum  values 
of  magnetism,  of  form  factor  p  =  18.5  (the  same  as  Fig.  63), 
and  ez  is  the  voltage  wave  induced  in  a  magnetic  circuit  having 
an  air-gap  of  1  per  cent,  of  its  length.  As  seen,  the  excessive 
peak  of  e  has  vanished,  and  ez  has  a  moderate  peak  only,  of  form 
factor  p  =  1.9. 

Even  a  much  smaller  air-gap  has  a  pronounced  effect  in  reducing 
the  voltage  peak.  Thus  curves  IV  and  V  show  the  m.m.fs.  of 
the  air-gap  and  of  the  total  magnetic  circuit,  respectively, 
when  containing  an  air-gap  of  one-thousandth  of  the  length  of 
magnetic  circuit.  e\  in  Fig.  70  then  shows  the  voltage  wave 
corresponding  to  V  in  Fig.  69 :  of  form  factor  p  =  7.4. 

Thus,  while  excessive  voltage  peaks  are  produced  in  a  highly 
saturated  closed  magnetic  circuit,  even  an  extremely  small  air- 
gap,  such  as  given  by  some  butt-joints,  materially  reduces  the 
peak:  from  form  factor  p  =  18.5  to  7.4  at  one-thousandth  gap 
length,  and  with  an  air-gap  of  1  per  cent,  length,  only  a  moderate 
peakedness  remains  at  the  highest  saturation,  while  at  lower 
saturation  the  voltage  wave  is  practically  a  sine. 

73.  Even  a  small  air-gap  in  the  magnetic  circuit  of  a  reactor 
greatly  reduces  the  wave-shape  distortion,  that  is,  makes  the 
voltage  wave  more  sinusoidal,  and  cuts  off  the  saturation  peak. 
The  latter,  however,  is  the  case  only  with  a  complete  air-gap. 
A  partial  air-gap  or  bridged  gap,  while  it  makes  the  wave  shape 


148 


ELECTRIC  CIRCUITS 


more  sinusoidal  elsewhere,  does  not  reduce  but  greatly  increases 
the  voltage  peak,  and  produces  excessive  peaks  even  below  satura- 
tion, with  a  sine  wave  of  current,  and  such  bridged  gaps  are,  there- 
fore, objectionable  with  series  reactors  in  high- voltage  circuits. 
In  shunt  reactors,  or  reactors  having  a  constant  sine  wave  of  im- 
pressed voltage,  the  bridged  gap  merely  produces  a  short  flat  zero 
of  the  current  wave,  thus  is  harmless,  and  for  these  purposes 
the  bridged  gap  reactance — shown  diagrammatically  in  Fig.  71 
— is  extensively  used,  due  to  its  constructive  advantages :  greater 


II-. 


7 


7 


Til 


FIG.  71. 

rigidity  or  structure  and,  therefore,  absence  of  noise,  and  reduced 
magnetic  stray  fields  and  eddy-current  losses  resulting  therefrom. 

Assuming  that  one-tenth  of  the  gap  is  bridged,  and  that  the 
length  of  the  gap  is  one  one-hundredth  that  of  the  entire  mag- 
netic circuit,  as  shown  diagrammatically  in  Fig.  71.  With  such 
a  bridged  gap,  with  all  but  the  lowest  m.m.fs.  the  narrow  iron 
bridges  of  the  gap  are  saturated,  thus  carry  the  flux  density 
S  +  H,  where  S  =  metallic  saturation  density,  =  20  kilolines 
per  cm.2  in  these  figures,  and  H  the  magnetizing  force  in  the  gap. 

For  one-tenth  of  the  gap,  the  flux  density  thus  is  H  +  S,  for 
the  other  nine-tenths,  it  is  H,  and  the  average  flux  density  in  the 
gap  thus  is 


SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION  149 

B  =  H  +  0.1  S  =  H  +  2, 
or,  if  p  =  bridged  fraction  of  gap, 

B  =  H  +  pS. 

Curve  II  in  Fig.  71  shows,  with  the  average  flux  density,  B,  as 
abscissae,  the  m.m.f.  required  by  the  gap, 

H  =  B  -  0.1  S 
=  B-2, 

while  curve  I  shows  the  m.m.f.  which  an  unbridged  gap  would 
require. 

Adding  to  the  ordinates  of  II  the  values  of  the  m.m.f.  required 
for  the  iron  part  of  the  magnetic  circuit,  or  the  other  99  per  cent., 
gives  as  curve  III  the  total  m.m.f.  of  the  reactor. 

The  lower  part  of  curve  III  is  once  more  shown,  with  five  times 
the  abscissae  B,  and  1000,  100  and  10  times,  respectively,  the 
ordinates  H,  as  IIIi.  III2,  His- 

74.  From  B  =  2  upward,  curve  III  is  practically  a  straight 
line,  and  plotting  herefrom  for  a  sine  wave  of  current,  /  and  thus 
m.m.f.,  H,  the  wave  of  magnetism,  B,  and  of  voltage,  e,  these  curves 
become  within  this  range  similar  to  a  sine  wave  as  shown  as  B  and 
e  in  Fig.  72.  Below  B  =  2,  however,  the  slope  of  the  B-H 
curve  and  with  this  their  wave  shapes  change  enormously.  The 
B  wave  becomes  practically  vertical,  that  is,  B  abruptly  reverses, 
and  corresponding  thereto,  the  voltage  abruptly  rises  to  an  ex- 
cessive peak  value,  that  is,  a  high  and  very  narrow  voltage  peak 
appears  on  top  of  the  otherwise  approximately  sine-shaped 
voltage  wave,  e. 

Choosing  the  same  value  as  in  Fig.  60,  B  =  15.4  or  beginning 
saturation,  as  the  maximum  value  of  flux  density:  at  this,  in  an 
entirely  closed  magnetic  circuit  the  voltage  peak  is  still  moderate. 
On  the  B-H  curve  III  of  Fig.  71,  the  flux  density,  B  =  15.4, 
requires  the  m.m.f.,  H  =  14.4  If  then  B  and  H  would  vary 
sinusoidally,  giving  a  sine  wave  of  voltage,  e0,  the  average  value 
of  this  voltage  wave,  60,  would  be  proportional  to  the  average  rate 

of  magnetic  change,  or  to  -jj  =  T^  =  1-07,  and  the  maximum 
value  of  the  sine  wave  of  voltage  would  be  ~  as  high,  or, 

7T    B  1.077T 

60  =  2  H  =  "2"  =  L68' 


150  ELECTRIC  CIRCUITS 

The  maximum  value  of  the  actual  voltage  curve,  6,  occurs  at 
the  moment  where  B  passes  through  zero,  and  is,  from  curve  IIIi, 

r£-|         290 
e  =     -FT      =  —^-  =  580. 


This,  then,  is  the  peak  voltage  of  the  actual  wave,  while,  if  it  were 
a  sine  wave,  with  the  same  maximum  magnetic  flux,  the  maxi- 
mum voltage  would  be  eQ  =  1.68. 

The  voltage  peak  produced  by  the  bridged  gap  and  the  form 
factor  thus  is 

e        580 


that  is,  345  times  higher  than  it  would  be  with  a  sine  wave. 

Obviously,  such  peak  can  hardly  ever  occur,  as  it  is  usually 
beyond  the  limit  of  the  available  supply  voltage.  It  thus  means, 
that  during  the  very  short  moment  of  time,  when  during  the 
current  reversal  the  flux  density  in  the  iron  bridge  of  the  gap 
changes  from  saturation  to  saturation  in  the  reverse  direction, 
a  voltage  peak  rises  up  to  the  limits  of  voltage  given  by  the  sup- 
ply system.  This  peak  is  so  narrow  that  even  the  oscillograph 
usually  does  not  completely  show  it. 

However,  such  practically  unlimited  peaks  occur  only  in  a 
perfectly  closed  magnetic  circuit,  containing  a  bridged  gap.  If, 
in  addition  to  the  bridged  gap  of  1  per  cent.,  an  unbridged  gap  of 
0.1  per  cent.  —  such  as  one  or  several  butt-joints  —  is  present, 
giving  the  B-H  curve  IV  of  Fig.  71,  the  voltage  peak  is  greatly 
reduced.  It  is 

TT  B       r   15.4       ,  K1 
60  =  2  H  =  2  1T95  =  L51' 


100 
hence,  the  relative  voltage  peak,  or  form  factor, 

p  =  —  =  6.6. 
e0 

That  is,  by  this  additional  gap  of  one  one-thousandth  of  the 
length  of  magnetic  circuit,  the  peak  voltage  is  reduced  from  345 
times  that  of  the  sine  wave,  to-  only  6.6  times,  or  to  less  than  2 
per  cent,  of  its  previous  value. 

As  seen  from  the  reasoning  in  paragraph  and  Fig.  67,  the 


SHAPING  OF  WAVES  BY  MAGNETIC  SATURATION  151 


peaked  wave  of  Fig.  72  contains  very  pronounced  harmonics  up 
to  about  the  701th,  which  at  60  cycles  of  fundamental  frequency, 
gives  frequencies  up  to  42,000,  or  well  within  the  range  of  the 
danger  frequencies  of  high-voltage  power  transformers,  that  is, 


B/'I 


FIG.  72. 

frequencies  with  which  the  high-voltage  coils  of  transformers,  as 
circuits  of  distributed  capacity,  can  resonate. 

75.  Magnetic  saturation,  and  closed  or  partly  closed  magnetic 
circuits  thus  are  a  likely  source  of  wave-shape  distortion,  resulting 
in  high  voltage  peaks,  and  where  they  are  liable  to  occur,  as  in 


152  ELECTRIC  CIRCUITS 

current  transformers,  series  transformers  at  open  secondary  cir- 
cuit, autotransformers  or  reactors,  etc.,  they  may  be  guarded 
against  by  using  a  small  air-gap  in  the  magnetic  circuit,  or  by 
providing  the  extra  insulation  required  to  stand  the  voltage,  and 
the  secondary  circuit,  even  if  of  an  effective  voltage  which  is  not 
dangerous  to  life  when  a  sine  wave,  should  be  carefully  handled 
as  the  voltage  peak  may  reach  values  which  are  dangerous  to 
life,  without  the  voltmeter — which  reads  the  effective  value — 
indicating  this. 

Inversely,  such  voltage  peaks  are  intentionally  provided  in 
some  series  autotransformers  for  the  operation  of  individual  arcs 
of  the  type,  in  which  slagging  and  consequent  failures  to  start  may 
occur,  due  to  a  high-resistance  slag  covering  the  electrode  tips. 
By  designing  the  autotransf  ormer  so  as  to  give  a  very  high  voltage 
peak  at  open  circuit — and  providing  in  the  apparatus  the  insula- 
tion capable  to  stand  this  voltage — reliability  of  starting  is  se- 
cured by  puncturing  any  non-conducting  slag  on  the  electrode 
tips,  by  the  voltage  peak. 

These  high  voltage  peaks,  produced  by  magnetic  saturation, 
etc.,  greatly  decrease  and  vanish  if  considerable  current  is  pro- 
duced by  them.  Thus,  when  the  secondary  of  a  closed  magnetic 
circuit  series  transformer  is  open,  at  magnetic  saturation,  a  high 
voltage  peak  appears;  with  increasing  load  on  the  secondary, 
however,  the  voltage  peak  drops  and  practically  disappears 
already  at  relatively  small  load.  Thus  such  arrangements  are 
suitable  for  producing  voltage  peaks  only  when  no  current  is 
required,  as  for  disruptive  effects,  or  only  very  small  currents. 


CHAPTER  IX 
WAVE  SCREENS.     EVEN  HARMONICS 

76.  The  elimination  of  voltage  and  current  distortion,  and 
production  of  sine  waves  from  any  kind  of  supply  wave,  that  is, 
the  reverse  procedure  from  that  discussed  in  the  preceding  chapter, 
is  accomplished  by  what  has  been  called  "wave  screens." 

Series  reactance  alone  acts  to  a  considerable  extent  as  wave 
screen,  by  consuming  voltage  proportional  to  the  frequency  and 
the  current,  and  thereby  reducing  the  harmonics  of  voltage  in 
the  rest  of  the  circuit  the  more,  the  higher  their  order. 

Let  the  voltage  impressed  upon  the  circuit  be  denoted  sym- 
bolically by 

-Z«.  (29) 

where  n  denotes  the  order  of  the  harmonic  of  absolute  numerical 
value  en. 

If,  then,  the  reactance  x  (at  fundamental  frequency)  is  inserted 
into  the  circuit  of  resistance,  r,  the  impedance  is 

z\   —   "v/r2  +  x2  for  the  fundamental  frequency,  and 
zn  =  \/r2~-f-  nV  for  the  nth  harmonic,  (30) 

and  the  current  thus  is 

i  =  e-  =  V          e      f-}  (31) 

or,  denoting 

r-  =  c,  (32) 

it  is 


153 


154 


ELECTRIC  CIRCUITS 


if  r  is  small  compared  with  x,  c2  is  negligible  compared  with  1, 
9,  25,  etc.,  and  it  is 


,   €5   .   €7   .  \ 

+5  +7  +•;.•  • 


that  is,  the  current,  i,  and  thus  the  voltage  across  the  resistance,  r, 
shows  the  harmonics  of  the  supply  voltage,  e,  reduced  in  propor- 
tion to  their  order,  n. 

Even  if  r  is  large  compared  with  x,  and  thus  c2>l,  finally  c2 
becomes  negligible  with  n2,  and  the  harmonics  decrease  with  their 
order. 

77.  The  screening  effect  of  the  series  reactance  is  increased  by 
shunting  a  capacity,  C,  beyond  the  inductance,  L,  that  is,  across 
the  resistance,  r,  as  shown  in  Fig.  73.  By  consuming  current 


_cmmr 


FIG.  73. 


FIG.  74. 


proportional  to  frequency  and  voltage,  the  condenser  shunts  the 
more  of  the  current  passing  through  the  reactance,  the  higher  the 
frequency,  and  thereby  still  further  reduces  the  higher  harmonics 
of  current  in  the  resistance,  r,  and  thus  of  voltage  across  this  re- 
sistance. Its  effect  is  limited,  however,  by  the  decreasing  voltage 
distortion  at  r  and  thus  at  the  condenser,  C. 

Thus  the  screening  effect  is  still  further  increased  by  inserting 
a  second  inductance,  L,  beyond  the  condenser,  C,  in  series  to  the 
resistance,  r,  as  shown  in  Fig.  74.  By  making  the  second  induct- 
ance equal  to  the  first  one,  and  making  the  condenser,  C,  of  the 
same  reactance,  for  the  fundamental  wave,  as  each  of  the  two 
inductances,  we  get  what  probably  is  the  most  effective  wave 
screen.  This  77-connection  or  resonating  circuit  will  be  discussed 
more  fully  in  Chapter  XIV,  in  its  feature  of  constant-potential 
constant-current  transformation. 

Under  the  condition,  that  the  two  inductive  reactances  and  the 


WAVE  SCREENS.     EVEN  HARMONICS  155 

capacity  reactance  are  equal,  the  equation  of  the  current  in  the 
resistance,  r,  is  (page  291),  for  the  nth  harmonic, 


/  _  Jw .  (VA\ 

xn(n2  -  2)  -  jr(n2  -  1) 


t  =  -°  X  (35) 

2  -  2)2  +  c2(n2  -  I)2 


or,  absolute, 


where  c  =  -  (36) 

If  c  is  small,  that  is,  r  small  compared  with  x,  the  current 
becomes 

t  i  =  xn(n>-2)  <37) 

or,  for  higher  values  of  n, 


that  is,  it  decreases  with  increasing  order  of  harmonic,  and  pro- 
portional to  the  cube  of  the  order  n,  thus  shows  an  extremely 
rapid  decrease. 

If  c  is  not  negligible,  the  denominator  in  (35)  is  larger,  and  it 
therefore,  still  smaller. 

As  illustration  may  be  shown  the  current,  i0,  and  thus  the  vol- 
tage, Co,  across  a  resistance,  r,  under  the  very  greatly  distorted 
and  peaked  voltage  of  Fig.  62: 

(a)  for  a  series  reactance,  z,  equal  to  r,  that  is,  c  =  1 ; 

(6)  for  the  complete  wave  screen  of  two  inductances  and  one 
capacity. 

It  is 
impressed  voltage, 

e  =  1.27  Co  {  li  +  0.9783  +  0.9355  +  0.8777  +  0.8009  +  0.713n 
+  0.617i3  +  0.517]5  +  0.416i7  +  0.315i9  +  0.1892J}. 

(a)  Reduction  factor  of  the  nth  harmonic, 
1  1 


hence, 

1.27 
ei   =  771^0   li  +  0.4423  +  0.2585  +  0.1757  +  0.1259  +  0.091  n 

+  0.067is  +  0.049i5  +  0.034i7  +  0.023i9  +  0.0132i}. 


156  ELECTRIC  CIRCUITS 

(b)  Reduction  factor  of  the  nth  harmonic, 

1_ 

/  n(n2  -  2)' 

hence, 

e2  =  1.27  e0  {li  +  0.0473  +  0.0085  +  0.0037  +  0.0019  +  0.001n|. 
That  is,  the  third  harmonic  is  reduced  to  less  than  5  per  cent., 
the  fifth  to  less  than  1  percent., and  the  higher  ones  are  practically 
entirely  absent. 

While  in  the  supply  voltage  wave,  e,  the  voltage  peak  (by  adding 
the  numerical  values  of  all  the  harmonics:  1  +  0.978  +  0.935  + 
.  .  .)  is  7.36  times  that  of  the  fundamental  wave,  it  is  reduced 
by  series  reactance  to  less  than  2.28  times  the  maximum  of  the 
fundamental  wave,  that  is,  very  greatly  reduced,  and  by  the 
complete  wave  screen  to  less  than  1.06  times  the  maximum  of 
the  fundamental.  That  is,  in  the  last  case 
the  voltage  is  practically  a  perfect  sine 
wave. 

78.  By  "wave  screens"  the  separation  of 
pulsating  currents  into  their  alternating  and 
their  continuous  component,  or  the  separa- 
tion of  complex  alternating  currents — and 
thus  voltages — into  their  constituent  har- 
monics can  be  accomplished,  and  inversely, 

the  combination  of  alternating  and  continuous  currents  or  vol- 
tages into  resultant  complex  alternating  or  pulsating  currents. 
The  simplest  arrangement  of  such  a  wave  screen  for  separating, 
or  combining  alternating  and  continuous  currents  into  pulsating 
ones,  is  the  combination,  in  shunt  with  each  other,  of  a  capacity, 
C,  and  an  inductance,  L,  as  shown  in  Fig.  75.  If,  then,  a  pulsating 
voltage,  e,  is  impressed  upon  the  system,  the  pulsating  current,  i, 
produced  by  it  divides,  as  the  continuous  component  can  not 
pass  through  the  condenser,  C,  and  the  alternating  component 
is  barred  by  the  inductance,  L,  the  more  completely,  the  higher 
this  inductance.  Thus  the  current,  i\t  in  the  apparatus,  A,  is  a 
true  alternating  current,  while  the  current,  IQ,  in  the  apparatus,  C, 
is  a  slightly  pulsating  direct  current. 

Inversely,  by  placing  a  source  of  alternating  voltage,  such  as 
an  alternator  or  the  secondary  of  a  transformer,  at  A,  and  a  source 
of  continuous  voltage,  such  as  a  storage  battery  or  direct-current 


WAVE  SCREENS.    EVEN  HARMONICS 


157 


generator,  at  C,  in  the  external  circuit  a  pulsating  voltage,  e,  and 
pulsating  current,  i,  result. 

If  the  capacity,  C,  is  so  large  as  to  practically  short-circuit  the 
alternating  voltage,  and  the  inductance,  L,  so  high  as  to  practically 
open-circuit  the  alternating  voltage,  the  separation — of  combi- 
nation— is  practically  complete,  and  independent  of  the  frequency 
of  the  alternating  wave. 

Wave  screens  based  on  resonance  for  a  definite  frequency  by 
series  connection  of  capacity  and  inductance,  can  be  used  to  sepa- 
rate the  current  of  this  frequency  from  a  complex  current  or 
voltage  wave,  such  as  those  given  in  Figs.  56  to  63,  and  thus  can 
be  used  for  separation  of 

complex    waves    into    their         c*||        ^(^MMMGG\  /^N 
components,   by  "harmonic 
analysis." 

Thus  in  Fig.  76,  if  the 
successive  capacities  and  in- 
ductances are  chosen  such 
that 


107T/L5    = 


2mrfLn 


1 


10  7T/C5' 
1 


(39) 


FIG.  76. 


where  /  =   frequency    of   the    fundamental    wave. 

Then,  through  any  of  the  branch  circuits  Cn,  Ln,  only  the  nth 

harmonic,  in,  can  pass  to  an  appreciable  extent. 

Such  resonant  wave  screen,  however,  has  the  serious  disadvan- 
tage to  require  very  high  constancy  of  /,  since  the  resonance  condi- 
tion between  Cn  and  Ln  depends  on  the  square  of  /, 


79.  Even  harmonics  are  produced  in  a  closed  magnetic  circuit 
by  the  superposition  of  a  continuous  current  upon  the  alternating 
wave.  With  an  alternating  sine  wave  impressed  upon  an  iron 
magnetic  circuit,  saturation,  or  in  general  the  lack  of  proportional- 


158 


ELECTRIC  CIRCUITS 


ity  between  magnetic  flux  and  m.m.f.,  produces  a  wave-shape  dis- 
tortion, that  is,  higher  harmonics,  of  voltage  with  a  sine  wave  of 
current,  of  current  with  a  sine  wave  of  impressed  voltage.  The 
constant  term  of  a  wave,  however,  is  the  first  even  harmonic,  and 
thus,  if  the  impressed  wave  comprises  a  fundamental  sine  and  a 


\ 


\ 


FIG.  77. 

constant  term,  the  former  gives  rise  to  the  odd  harmonics,  the 
latter  to  the  even  harmonics. 

Let,  then,  on  the  alternating  sine  wave  of  impressed  voltage  a 
continuous  current  by  superimposed.  The  magnetic  flux  then 
oscillates  sinusoidally,  not  between  equal  and  opposite  values, 
but  between  two  unequal  values,  which  may  be  of  the  same,  or  of 
opposite  signs.  That  is,  it  performs  an  unsymmetrical  magnetic 
cycle.  Neglecting  again  hysteresis,  that  is,  assuming  the  rising 


WAVE  SCREENS.     EVEN  HARMONICS  159 

and  the  decreasing  magnetization  curve  as  coincident — which  is 
permissible  as  approximation,  since  the  hysteresis  contributes 
little  to  distortion — and  choosing  the  same  magnetization  curve 
as  in  the  preceding,  curve  I  in  Fig.  64,  we  may  as  an  instance  con- 
sider a  sinusoidal  magnetic  pulsation  between  the  limits  +15.4 
and  +19.7,  corresponding  to  a  variation  of  the  m.m.f.  between 
H  =  +10  and  H  =  +100. 

Fig.  77  then  gives,  as  curve  B,  the  sinusoidally  pulsating  mag- 
netic flux  density.  Taking  from  curve  /,  Fig.  64,  the  values  of  H 
corresponding  to  the  values  in  curve  B,  Fig.  77,  gives  curve  H. 
This,  resolved  (" Engineering  Mathematics,"  paragraph  92) 
gives  the  constant  term  iQ  =  36,  and  the  alternating  current,  i. 
The  latter  is  unsymmetrical,  having  one  short  half-wave  of  a  peak 
value  64,  and  one  long  half-wave  of  maximum  value  26.  It  thus 
resolves  into  the  odd  harmonics,  ii,  alternating  between  ±  45,  and 
the  even  harmonics,  mainly  the  second  harmonic,  alternating 
between  maximum  values  +18  and  —15.  ii  is  peaked  with  flat 
zero,  thus  showing  a  third  harmonic,  which  is  separated  as  i3,  and 
i2  is  unsymmetrical,  showing  further  even  harmonics,  which  are 
separated  as  z'4,  but  are  rather  small. 

Thus  the  pulsating  exciting  current  of  the  sinusoidally  varying 
unidirectional  magnetic  flux 

B  =  17.55  +  2.15  cos  4 
is  given  by 

H  =  36  +  37cos0  +  16.5cos20  +  8cos30  +  2  cos40  +  .  . 

Instead  of  superimposing  a  direct  current  upon  an  alternating 
wave,  as  by  connecting  in  series  an  alternator  and  a  direct-current 
generator  or  storage  battery,  two  separate  coils  can  be  used  on  the 
magnetic  circuit,  one  energized  by  an  alternating  impressed  vol- 
tage, the  other  by  a  direct  current.  A  high  inductive  reactance 
would  then  be  connected  in  the  latter  circuit,  to  eliminate  the 
current  pulsation  which  would  be  caused  by  the  alternating  vol- 
tage induced  in  this  coil. 

Connecting  two  such  magnetic  circuits  with  their  direct-current 
magnetizing  coils  in  series,  but  in  opposition  (without  the  use  of  a 
series  reactance)  eliminates  the  induced  fundamental  wave,  but 
leaves  the  second  harmonic  in  the  direct-current  circuit,  which 
thus  can  be  separated.  Numerous  arrangements  can  then  be  de- 
vised by  two  magnet  cores  energized  by  separate  alternating- 


160  ELECTRIC  CIRCUITS 

current  exciting  coils  and  saturated  by  one  common  direct-current 
exciting  coil,  surrounding  both  cores,  or  their  common  return,  etc. 
80.  The  preceding  may  illustrate  some  of  the  numerous  wave- 
shape distortions  which  are  met  in  electrical  engineering,  their 
characteristics,  origin,  effects,  use  and  danger.  Numerous  other 
wave  distortions,  such  as  those  produced  by  arcs,  by  unidirec- 
tional conductors,  by  dielectric  effects  such  as  corona,  by  Y  con- 
nection of  transformers  for  reactors,  by  electrolytic  polarization, 
by  pulsating  resistance  or  reactance,  etc.,  are  discussed  in  other 
chapters  or  may  be  studied  in  a  similar  manner. 


CHAPTER  X 

INSTABILITY  OF  CIRCUITS :  THE  ARC 
A.  General 

81.  During  the  earlier  days  of  electrical  engineering  practi- 
cally all  theoretical  investigations  were  limited  to  circuits  in  stable 
or  stationary  condition,  and  where  phenomena  of  instability 
occurred,  and  made  themselves  felt  as  disturbances  or  troubles  in 
electric  circuits,  they  either  remained  ununderstood  or  the  theo- 
retical study  was  limited  to  the  specific  phenomenon,  as  in  the 
case  of  lightning,  dropping  out  of  step  of  induction  motors,  hunt- 
ing of  synchronous  machines,  etc.,  or,  as  in  the  design  of  arc  lamps 
and  arc-lighting  machinery,  the  opinion  prevailed  that  theoretical 
calculations  are  impossible  and  only  design  by  trying,  based  on 
practical  experience,  feasible. 

The  first  class  of  unstable  phenomena,  which  was  systemat- 
ically investigated,  were  the  transients,  and  even  today  it  is  ques- 
tionable whether  a  systematic  theoretical  classification  and  in- 
vestigation of  the  conditions  of  instability  in  electric  circuits  is 
yet  feasible.  Only  a  preliminary  classification  and  discussion 
of  such  phenomena  shall  be  attempted  in  the  following. 

Three  main  types  of  instability  in  electric  systems  may  be 
distinguished : 

I.  The  transients  of  readjustment  to  changed  circuit   con- 
ditions. 

II.  Unstable  electrical  equilibrium,  that  is,  the  condition  in 
which  the  effect  of  a  cause  increases  the  cause. 

III.  Permanent  instability  resulting  from  a  combination  of 
circuit  constants  which  can  not  coexist. 

I.  TRANSIENTS 

82.  Transients  are  the  phenomena  by  which,  at  the  change  of 
circuit  conditions,   current,   voltage,  etc.,  readjust  themselves 
from  the  values  corresponding  to  the  previous  condition  to  the 
values  corresponding  to  the  new  condition  of  the  circuit.     For  in- 

11  161 


162  ELECTRIC  CIRCUITS 

stance,  if  a  switch  is  closed,  and  thereby  a  load  put  on  the  circuit, 
the  current  can  not  instantly  increase  to  the  value  corresponding 
to  the  increased  load,  but  some  time  elapses,  during  which  the 
increase  of  the  stored  magnetic  energy  corresponding  to  the  in- 
creased current,  is  brought  about.  Or,  if  a  motor  switch  is  closed, 
a  period  of  acceleration  intervenes  before  the  flow  of  current  be- 
comes stationary,  etc. 

The  characteristic  of  transients  therefore  is,  as  implied  in  the 
term,  that  they  are  of  limited,  usually  very  short  duration,  inter- 
vening between  two  periods  of  stationary  conditions. 

Considerable  theoretical  work  has  been  done,  more  or  less 
systematically,  on  transients,  and  a  great  mass  of  information  is 
thus  available  in  the  literature.  These  transients  are  more  ex- 
tensively treated  in  "Theory  and  Calculation  of  Transient  Elec- 
tric Phenomena  and  Oscillations,"  and  in  " Electric  Discharges, 
Waves  and  Impulses,"  and  therefore  will  be  omitted  in  the  fol- 
lowing. However,  to  some  extent,  the  transients  of  our  theoret- 
ical literature,  still  are  those  of  the  "phantom  circuit,"  that  is, 
a  circuit  in  which  the  constants  r,  L,  C,  g,  are  assumed  as  constant. 
The  effect  of  the  variation  of  constants,  as  found  more  or  less  in 
actual  circuits :  the  change  of  L  with  the  current  in  circuits  con- 
taining iron;  the  change  of  C  and  g  with  the  voltage  (corona,  etc.) ; 
the  change  of  r  and  g  with  the  frequency,  etc.,  has  been  studied  to 
a  limited  extent  only,  and  in  specific  cases. 

In  the  application  of  the  theory  of  transients  to  actual  electric 
circuits,  considerable  judgment  thus  is  often  necessary  to  allow 
and  correct  for  these  "secondary"  phenomena  which  are  not  in- 
cluded in  the  theoretical  equations. 

Especially  deficient  is  our  knowledge  of  the  conditions  under 
which  the  attenuation  constant  of  the  transient  becomes  zero 
or  negative,  and  the  transient  thereby  becomes  permanent,  or 
becomes  a  cumulative  surge,  and  the  phenomenon  thereby  one  of 
unstable  equilibrium. 

II.  UNSTABLE  ELECTRICAL  EQUILIBRIUM 

83.  If  the  effect  brought  about  by  a  cause  is  such  as  to  oppose 
or  reduce  the  cause,  the  effect  must  limit  itself  and  stability  be 
finally  reached.  If,  however,  the  effect  brought  about  by  a  cause 
increases  the  cause,  the  effect  continues  with  increasing  intensity, 
that  is,  instability  results. 


INSTABILITY  OF  CIRCUITS 


163 


This  applies  not  to  electrical  phenomena  alone,  but  equally  to 
all  other  phenomena. 

Instability -of  an  electric  circuit  may  assume  three  different 
forms : 

1.  Instability  leading  up  to  stable  conditions. 

For  instance,  in  a  pyroelectric  conductor  of  the  volt-ampere 
characteristic  given  in  Fig.  78,  at  the  impressed  voltage,  e0, 
three  different  values  of  current  are  possible:  i\t  iz  and  i3.  i\  and 
2*3  are  stable,  i2  unstable.  That  is,  at  current,  i^  passing  through 
the  conductor  under  the  constant  impressed  voltage,  eQ,  a  mo- 
mentary increase  of  current  would  give  an  excess  voltage  beyond 
that  required  by  the  conductor,  thereby  increase  the  current  still 


\ 


FIG.  78. 

further,  and  with  increasing  rapidity  the  current  would  rise,  until 
it  becomes  stable  at  the  value,  z'3.  Or,  a  momentary  decrease  of 
current,  by  requiring  a  higher  voltage  than  available,  would 
further  decrease  the  current,  and  with  increasing  rapidity  the 
current  would  decrease  to  the  stable  value,  i\. 

2.  Instability  putting  the  circuit  out  of  service. 

An  instance  is  the  arc  on  constant-potential  supply.  With  the 
volt-ampere  characteristic  of  the  arc  shown  as  A,  in  Fig.  79,  a 
current  of  4  amp.  would  require  80  volts  across  the  arc  terminals. 
At  a  constant  impressed  voltage  of  80,  the  current  could  not  re- 
main at  4  amp.,  but  the  current  would  either  decrease  with  in- 
creasing rapidity,  until  the  arc  goes  out,  or  the  current  would  in- 


164 


ELECTRIC  CIRCUITS 


crease  with  increasing  rapidity,  up  to  short-circuit,  that  is,  until 
the  supply  source  limits  the  current. 

3.  Instability  leading  again  to  instability,  and  thus  periodically 
repeating  the  phenomena. 

For  instance,  if  an  arc  of  the  volt-ampere  characteristic,  A, 
in  Fig.  79  is  operated  in  a  constant-current  circuit  of  sufficiently 
high  direct  voltage  to  restart  the  arc  when  it  goes  out,  and  the  arc 


80_ 
_70- 


-50. 
_40_ 


20. 


_JOJ 


FIG.  79. 

is  shunted  by  a  condenser,  the  condenser  makes  the  arc  unstable 
and  puts  it  out;  the  available  supply  voltage,  however,  starts 
it  again,  and  so  periodically  the  arc  starts  and  extinguishes,  as 
an  "oscillating  arc." 

84.  There  are  certain  circuit  elements  which  tend  to  produce 
instability,  such  as  arcs,  pyroelectric  conductors,  condensers, 
induction  and  synchronous  motors,  etc.,  and  their  recognition 
therefore  is  of  great  importance  to  the  engineer,  in  guarding 


INSTABILITY  OF  CIRCUITS  165 

against  instability.  Whether  instability  results,  and  what  form 
it  assumes,  depends,  however,  not  only  on  the  "exciting  element," 
as  we  may  call  the  cause  of  the  instability,  but  on  all  the  elements 
of  the  circuit.  Thus  an  arc  is  unstable,  form  (2),  on  constant- 
voltage  supply  at  its  terminals;  it  is  stable  on  constant-current 
supply.  But  when  shunted  by  a  condenser,  it  becomes  un- 
stable on  constant  current,  and  the  instability  may  be  form  (2) 
or  form  (3),  depending  on  the  available  voltage.  With  a  resist- 
ance, r,  of  volt-ampere  characteristic  ir  shown  as  B,  in  Fig.  79, 
the  arc  is  stable  on  constant- voltage  supply  for  currents  above  IQ 
=  3  amp.,  unstable  below  3  amp.,  and  therefore,  with  a  constant- 
supply  voltage,  e0,  two  current  values,  i\  and  i2,  exist,  of  which  the 
former  one  is  stable,  the  latter  one  unstable.  That  is,  current, 
*2,  can  not  persist,  but  the  current  either  runs  up  to  ii  and  the  arc 
then  gets  stable  (form  1),  or  the  current  decreases  and  the  arc 
goes  out,  instability  form  (2). 

Thus  it  is  not  feasible  to  separately  discuss  the  different  forms 
of  instability,  but  usually  all  three  may  occur,  under  different 
circuit  conditions. 

The  electric  arc  is  the  most  frequent  and  most  serious  cause  of 
instability  of  electric  circuits,  and  therefore  should  first  be  sus- 
pected, especially  if  the  instability  assumes  the  form  of  high- 
frequency  disturbances  or  abrupt  changes  of  current  or  voltage, 
such  as  is  shown  for  instance  in  the  oscillograms,  Figs.  80  and  81. 

Somewhat  similar  effects  of  instability  are  produced  by  pyro- 
electric  conductors. 

Induction  motors  and  synchronous  motors  may  show  instability 
of  speed :  dropping  out  of  step,  etc. 

III.  PERMANENT  INSTABILITY 

86.  If  the  constants  of  an  electric  circuit,  as  resistance,  in- 
ductance, capacity,  disruptive  strength,  voltage,  speed,  etc., 
have  values,  which  can  not  coexist,  the  circuit  is  unstable,  and 
remains  so  as  long  as  these  constants  remain  unchanged. 

Case  (3)  of  II,  unstable  equilibrium,  to  some  extent  may  be 
considered  as  belonging  in  this  class. 

The  most  interesting  class  in  this  group  of  unstable  electric 
systems  are  the  oscillations  resulting  sometimes  from  a  change 
of  circuit  conditions  (switching,  change  of  load,  etc.),  which  con- 
tinue indefinitely  with  constant  intensity,  or  which  steadily 
increase  in  intensity,  and  may  thus  be  called  permanent  and 


166  ELECTRIC  CIRCUITS 

cumulative  surges,  hunting,  etc.  They  may  be  considered  as 
transients  in  which  the  attenuation  constant  is  zero  or  negative. 

In  the  transient  resulting  from  a  change  of  circuit  conditions, 
the  energy  which  represents  the  difference  of  stored  energy  of  the 
circuit  before  and  after  the  change  of  circuit  condition,  is  dissi- 
pated by  the  energy  loss  in  the  circuit.  As  energy  losses  always 
occur,  the  intensity  of  a  true  transient  thus  must  always  be  a 
maximum  at  the  beginning,  and  steadily  decrease  to  zero  or  per- 
manent condition.  An  oscillation  of  constant  intensity,  or  of 
increasing  intensity,  thus  is  possible  only  by  an  energy  supply 
to  the  oscillating  system  brought  about  by  the  oscillation.  If 
this  energy  supply  is  equal  to  the  energy  dissipation,  constancy 
of  the  phenomenon  results.  If  the  energy  supply  is  greater  than 
the  energy  dissipation,  the  oscillation  is  cumulative,  and  steadily 
increases  until  self-destruction  of  the  system  results,  or  the  in- 
creasing energy  loss  becomes  equal  to  the  energy  supply,  and  a 
stationary  condition  of  oscillation  results.  The  mechanism  of 
this  energy  supply  to  an  oscillating  system  from  a  source  of  energy 
differing  in  frequency  from  that  of  the  oscillation  is  still  practi- 
cally unknown,  and  very  little  investigating  work  has  been  done 
to  clear  up  the  phenomenon.  It  is  not  even  generally  realized 
that  the  phenomenon  of  a  permanent  or  cumulative  line  surge 
involves  an  energy  supply  or  energy  transformation  of  a  fre- 
quency equal  to  that  of  the  oscillation. 

Possibly  the  oldest  and  best-known  instance  of  such  cumulative 
oscillation  is  the  hunting  of  synchronous  machines. 

Cumulative  oscillations  between  electromagnetic  and  electro- 
static energy  have  been  observed  by  their  destructive  effects  in 
high- voltage  electric  circuits  on  transformers  and  other  apparatus, 
and  have  been,  in  a  number  of  instances  where  their  frequency 
was  sufficiently  low,  recorded  by  the  oscillograph.  They  obvi- 
ously are  the  most  dangerous  phenomena  in  high-voltage  electric 
circuits.  Relatively  little  exact  knowledge  exists  of  their  origin. 
Usually — if  not  always — an  arc  somewhere  in  the  system  is 
instrumental  in  the  energy  supply  which  maintains  the  oscilla- 
tion. In  some  instances,  as  in  wireless  telegraphy,  they  have 
found  industrial  application.  A  systematic  theoretical  investiga- 
tion of  these  cumulative  electrical  oscillations  probably  is  one  of 
the  most  important  problems  before  the  electrical  engineer  today. 

The  general  nature  of  these  permanent  and  cumulative  oscilla- 
tions and  their  origin  by  oscillating  energy  supply  from  the  transi- 


INSTABILITY  OF  CIRCUITS 


167 


ent  of  a  change  of  circuit  condition,  is  best  illustrated  by  the  in- 
stance of  the  hunting  of  synchronous  machines,  and  this  will, 
therefore,  be  investigated  somewhat  more  in  detail. 

B.  The  Arc  as  Unstable  Conductor 

86.  The  instability  of  the  arc  is  the  result  of  its  dropping  volt- 
ampere  characteristic,  as  discussed  in  paragraphs  18  to  27  of  the 


100_ 


\ 


\ 


\ 


"o     ARC  ON 
-CONSTANT  VOLTAGE 
SUPPLY 


1.0     L5     2.0     2.5     3.0     3.5     4.0     4.5     5.0     5.5     6.0     6.5    7.0 


-.2 


FIG.  82. 


chapter  on  "Electric  Conductors."     As  shown  there,  the  arc  is 
always  unstable  on  constant  voltage  impressed  upon  it.     Series 


168  ELECTRIC  CIRCUITS 

resistance  or  reactance  produces  stability  for  currents  above  a 
certain  critical  value  of  current,  IQ.  Such  curves,  giving  the  vol- 
tage consumed  by  the  arc  and  its  series  resistance  as  function  of 
the  current,  thus  may  be  termed  stability  curves  of  the  arc.  Their 
minimum  values,  that  is,  the  stability  limits  corresponding  to  the 
different  resistances,  give  the  stability  characteristic  of  the  arc. 
The  equations  of  the  arc,  and  of  its  stability  curves  and  stability 
characteristic,  are  given  in  paragraphs  22  and  23  of  the  chapter  on 
"Electric  Conductors." 

Let,  in  Fig.  82,  A  present  the  volt-ampere  characteristic  of  an 
arc,  given  approximately  by  the  equation 


where 

*  -  4=  (2) 

Vi 

is  the  stream  voltage,  that  is,  voltage  consumed  by  the  arc  stream. 
Fig.  82  is  drawn  with  the  constants, 

a  =  35, 
c  =  51, 
I  =  1.8, 
5  =  0.8, 
hence, 


Assuming  this  arc  is  operated  from  a  circuit  of  constant-  voltage 
supply, 

E  =  150  volts, 
through  a  resistance,  r0 

The  voltage  consumed  by  the  resistance,  r0,  then  is 

e2  =  r0i,  (3) 

and  the  voltage  available  for  the  arc  thus 

ei  =  E  -  TV  (4) 

Lines  B,  C  and  D  of  Fig.  82  give  e\,  for  the  values  of  resistance, 

r0  =  20  ohms  (B) 
=  10  ohms  (C) 
=  13  ohms  (D). 


INSTABILITY  OF  CIRCUITS  169 

As  seen,  line  B  does  not  intersect  the  volt-ampere  characteris- 
tic, A,  of  the  arc,  that  is,  with  20  ohms  resistance  in  series,  this  I  — 
2.5  cm.  arc  can  not  be  operated  from  E  =  150  volt  supply. 

Line  C  intersects  A  at  a  and  b}  i  =  6.1  and  1.9  amp.  respect- 
ively. 

At  a,  i  =  6.1  amp.,  the  arc  is  stable; 

At  b,  i  =  1.9  amp.,  the  arc  is  unstable; 

for  the  reasons  discussed  before  :  an  increase  of  current  decreases 
the  voltage  consumed  by  the  circuit,  e  +  e2,  and  thus  still  further 
increases  the  current,  and  inversely.  Thus  the  arc  either  goes  out, 
or  the  current  runs  up  to  i  =  6.1  amp.,  where  the  arc  gets  stable. 

Line  D  is  drawn  tangent  to  A,  and  the  contact  point,  c,  thus 
gives  the  minimum  current,  i  =  3.05  amp.,  of  operation  of  the  arc 
on  E  =  150  volts,  that  is,  the  value  of  current  or  of  series  resist- 
ance, at  which  the  arc  ceases  to  be  stable  :  a  point  of  the  stability 
characteristic,  St  of  the  arc. 

This  stability  characteristic  is  determined  by  the  condition 


where 

e0  =  e  +  TQI  (6) 


this  gives 
and 


=  a  +  —/=.  +  rQi, 


r"  =  ^  =  ^  (7) 


(8) 


=  a  +  1.5  6 


a  +  1.5  e\ 


as  the  equation  of  the  stability  characteristic  of  the  arc  on  a  con- 
stant-voltage circuit. 

87.  In  general,  the  condition  of  stability  of  a  circuit  operated  on 
constant-  voltage  supply,  is 


where  e  is  the  voltage  consumed  by  the  current,  i,  in  the  circuit. 

The  ratio  of  the  change  of  voltage,  de,  as  fraction  of  the  total 
voltage,  e,  brought  about  by  a  change  of  current,  di,  as  fraction  of 


170 


ELECTRIC  CIRCUITS 


the  total  current,  i,  thus  may.be  called  the  stability  coefficient  of  the 
circuit, 

de 


de 

e 
i 

In  a  circuit  of  constant  resistance,  r,  it  is 


(10) 


de 


hence, 


=  1, 


that  is,  the  stability  coefficient  of  a  circuit  of  constant  resistance, 
r,  is  unity. 

In  general,  if  the  effective  resistance,  r,  is  not  constant,  but  varies 
with  the  current,  i,  it  is 

e  =  ri, 

de  .  dr 


hence,  the  stability  coefficient 


dr 


5  =  1  -f  _ 


(11) 


thus  in  a  circuit,  in  which  the  resistance  increases  with  the  current, 
the  stability  coefficient  is  greater  than  1.  Such  is  that  of  a  con- 
ductor with  positive  temperature  coefficient  of  resistance,  in 
which  the  temperature  rise  due  to  the  increase  of  current  increases 
the  resistance.  A  conductor  with  negative  temperature  coeffici- 
ent of  resistance  gives  a  stability  coefficient  less  than  1,  but  as  long 
as  6  is  still  positive,  that  is,  the  decrease  of  resistance  slower  than 
the  increase  of  current,  the  circuit  is  stable. 


6  >  0 


(12) 


INSTABILITY  OF  CIRCUITS  171 

is  the  condition  of  stability  of  a  circuit  on  constant-  voltage  supply, 
and 

d  <  0  (13) 

is  the  condition  of  instability,  and 

6  =  0  (14) 

thus  gives  the  stability  characteristic  of  the  circuit. 
In  the  arc, 


the  stability  coefficient  is,  by  (10), 


that  is,  equals  half  the  stream  voltage,  ^  >   divided  by    the    arc 

voltage,  e. 

Or,  substituting  for  e  in  (15),  and  rearranging, 

,=  __   _J_ 

(16) 


2(1  +  0.2625      O 
in  Fig.  82. 

Fori  =  0,  it  is  6=  -0.5; 
i  —  oo ,  it  is  6  =  0. 

The  stability  coefficient  of  the  arc  having  the  volt-ampere 
characteristic,  A,  in  Fig.  82  is  shown  as  F  in  Fig.  82. 

88.  On  constant- voltage  supply,  E  =  150  volts,  the  arc  having 
the  characteristic,  A,  Fig.  82,  can  not  be  operated  at  less  than  3.05 
amperes.  At  i  =  3.05  is  its  stability  limit,  that  is,  the  stability 
coefficient  of  arc  plus  series  resistance,  r0,  required  to  give  150 
volts,  changes  from  negative  for  lower  currents,  to  positive  for 
higher  currents. 

The  stability  coefficient  of  such  arcs,  operated  on  constant- 
voltage  supply  through  various  amounts  of  series  resistance,  r<>, 
then  would  be  given  by 

de. 

di 

do  =  — , 
Co 

i 
where 


172  ELECTRIC  CIRCUITS 


e0  =  a  +  — 7=  +  r0i  (17) 


and  the  resistance  r0  chosen  so  as  to  give 

e\  =  150  volts, 
from  (17)  follows, 


and,  substituting  from  (17), 

vi  0     —     ^0  Cv    """"          7— 

V? 
gives 

1.56 

0  +   V^  (18) 


or, 


So  =  1  -  -  (19) 


where  e0  is  the  supply  voltage,  e'Q  the  voltage  given  by  the  stability 
characteristic,  S. 

60,  the  stability  characteristic  of  the  arc,  A,  on  E  =  150  volt 
constant-potential  supply,  is  given  as  curve,  G,  in  Fig.  82.  As 
seen,  it  passes  from  negative  —  instability  —  to  positive  —  stability 
—  at  the  point,  k,  corresponding  to  c  and  h  on  the  other  curves. 

89.  On  a  constant-current  supply,  an  arc  is  inherently  stable. 
Instability,  however,  may  result  by  shunting  the  arc  by  a  resist- 
ance, ri.  Thus  in  Fig.  83,  let  /  =  5  amp.  be  the  constant  supply 
current.  The  volt-ampere  characteristic  of  the  arc  is  given  by  A, 
and  shows  that  on  this  5-amp.  circuit,  the  arc  consumes  94  volts, 
point  d. 

Let  now  the  arc  be  shunted  by  resistance,  r\.  If  e  =  voltage 
consumed  by  the  arc,  the  current  shunted  by  the  resistance,  r\t  is 

*i  =  ^  (2°) 

and  the  current  available  for  the  arc  thus  is 

i  =  /  -  ii  (21) 


or 

e  =  n(/  -  iX  (22) 


INSTABILITY  OF  CIRCUITS 


173 


Curves  B,  C  and  D  of  Fig.  83  show  the  values  of  equation  (22) 


for 


7*1  =  32  ohms:  line  B 
=  48  ohms:  line  C 
=  40.8  ohms:  line  D. 


ON 

CONSTANT  CURRENT 
SUPPLY 


FIG.  83. 

Line  B  does  not  intersect  the  arc  characteristic,  A,  that  is,  with 
a  resistance  as  low  as  r\  —  32,  no  arc  can  be  maintained  on  the 
5-amp.  constant-current  circuit. 

Line  C  intersects  A  at  two  points: 

(a)  i  =  2.55  amp.,  e  =  118  volts,  stable  condition; 

(b)  i  =  0.55  amp.,  e  =  214  volts,  unstable  condition. 

Line  D  is  drawn  tangent  to  A,  touches  at  c:  i  —  1.4  amp., 


174  ELECTRIC  CIRCUITS 

e  =  148  volts,  the  limit  of  stability.  At  7  =  5  amp.,  the  point  h, 
&te  =  148  volts,  thus  gives  the  voltage  consumed  by  an  arc  when 
by  shunting  it  with  a  resistance  the  stability  limit  is  reached. 

Drawing  then  from  the  different  points  of  the  abscissae,  z, 
tangents  on  A,  and  transferring  their  contact  points,  c,  6,  to  the 
abscissae,  from  which  the  tangent  is  drawn,  gives  the  points  h, 
g,  of  the  constant-current  stability  characteristic  of  the  arc,  that 
is,  the  curve  of  arc  voltages  in  a  constant-current  circuit,  7,  when 
by  shunting  the  arc  with  a  resistance,  n,  consuming  current,  ii, 
the  stability  limit  of  the  arc  with  current  i  =  I—ii  is  reached. 

P  then  gives  the  curve  of  the  arc  currents,  i,  corresponding  to 
the  arc  voltage,  e,  of  curve  Q,  for  the  different  values  of  the  con- 
stant-circuit current,  7. 

The  equations  of  Q  and  P  are  derived  as  follows  : 

The  stability  limit,  point  c,  corresponding  to  circuit  current,  7, 
as  given  by 

de 


where  e  =  arc  voltage,  and  i  =  arc  current. 
Or, 

,,  *-a77r  (23) 

It  is,  however, 

e  =  a  -\  --  /= 
yi 
and 


From  these  three  equations  follows,  by  eliminating  r\  and  i  or  e, 

Q, 

I  -  '-f^  (24) 

P, 


These  curves  are  of  lesser  interest  than  the  constant-voltage 
stability  curve  of  the  arc,  S  in  Fig.  82. 

It  is  interesting  to  note,  that  the  resistance,  r\  (23),  which 
makes  an  arc  unstable  as  shunting  resistance  in  a  constant- 
current  circuit,  has  the  same  value  as  the  resistance,  r0,  (7),  which 


INSTABILITY  OF  CIRCUITS 


175 


as  series  resistance  makes  it  unstable  in  a  constant- voltage  supply 
circuit. 

90.  Due  to  the  dropping  volt-ampere  characteristic,  two  arcs 
can  not  be  operated  in  parallel,  unless  at  least  one  of  them  has  a 
sufficiently  high  resistance  in  series. 


PARALLEL 

OPERATION 

OF  ARCS 


.5   1.0  15  2.0  25  30  3.5  40  45  50  5.5  60  65  7.0 


FlG.  84. 


Let,  as  shown  in  Fig.  84,  two  arcs  be  connected  in  parallel  into 
the  circuit  of  a  constant  current 

7  =  6  amp. 

Assume  at  first  both  arcs  of  the  same  length  and  same  electrode 
material,  that  is,  the  same  volt-ampere  characteristic. 


176  ELECTRIC  CIRCUITS 

Let  i  =  current  in  the  first  arc,  thus  i'  =  I  —  i  =  current  in 
the  second  arc. 

The  volt-ampere  characteristic  of  the  first  arc,  then,  is  given  by 
A  in  Fig.  84,  that  of  the  second  arc  by  A'. 

As  the  two  parallel  arcs  must  have  the  same  voltage,  the  oper- 
ating point  is  the  point,  a,  of  the  intersection  of  A  and  A'  in 
Fig.  84. 

The  arcs  thus  would  divide  the  current,  each  operating  at  3 
amp. 

However,  the  operation  is  unstable :  if  the  first  arc  should  take  a 
little  more  current,  its  voltage  decreases,  on  curve  A,  that  of  the 
second  arc  increases,  on  A',  due  to  the  decrease  of  its  current,  and 
the  first  arc  thus  takes  still  more  current,  thus  robs  the  second  arc, 
the  latter  goes  out  and  only  one  arc  continues. 

Thus  two  arcs  in  parallel  are  unstable,  and  one  of  them  goes  out, 
only  one  persists. 

Suppose  now  a  resistance  of 

r  =  30  ohms 

is  connected  in  series  with  each  of  the  two  arcs,  as  shown  in 
Fig.  84. 

The  volt-ampere  characteristics  of  arc  plus  resistance,  r,  then, 
are  given  by  curves  B  and  B'. 

These  intersect  in  three  points:  b,  g  and  h. 

Of  these,  point  b  is  stable :  an  increase  of  the  current  in  one  of 
the  arcs,  and  corresponding  decrease  in  the  other,  increases  the 
voltage  consumed  by  the  circuit  of  the  former,  decreases  that  con- 
sumed by  the  circuit  of  the  latter,  and  thus  checks  itself. 

The  points  g  and  h,  however,  are  unstable. 

At  6,  stable  condition,  the  characteristics,  B  and  B',  are  rising; 
at  a,  unstable  condition,  the  characteristics,  A  and  A',  are  drop- 
ping, and  the  stability  limit  is  at  that  value  of  resistance,  r,  at 
which  the  circuit  characteristics  plus  resistance,  are  horizontal, 
the  point  c,  where  the  characteristics,  C  and  C",  touch  each  other. 

c  is  the  stability  limit  of  C  or  C',  thus  a  point  of  the  stability 
characteristic  of  either  arc,  or  given  by  the  equation 

1.56 

e  =  a  H T=. 

V  ^ 

Fig.  85  shows  the  case  of  two  parallel  arcs,  which  are  not  equal 
and  do  not  have  equal  resistances,  r,  in  series,  one  being  a  long  arc, 


INSTABILITY  OF  CIRCUITS 


177 


having  no  resistance  in  series,  the  other  a  short  arc  with  a  resist- 
ance r  =  40  ohms  in  series. 

The  volt-ampere  characteristic  of  the  long  arc  is  given  by  A, 
that  of  the  short  arc  by  B,  and  that  of  the  short  arc  plus  resistance, 
r,byC. 

A  and  C  intersect  at  three  points,  a,  6  and  c.  Of  these,  only  the 
point  a  is  stable,  as  any  change  of  current  from  this  point  limits 


280 
2V 
240. 
220_ 
200. 
ISO. 
160 
.140- 
120- 
100- 
_80_ 


-40- 
20_ 


5      10     15    20    2.5    30    35     40    45     50    5  5    6  0     65    1 0 


PARALLEL  OPERATION 
OF  ARCS 


FIG.  85. 

itself;  b  and  c,  however,  are  unstable.  Thus,  at  the  latter  points, 
the  arcs  can  not  run,  but  the  current  changes  until  either  one 
arc  has  gone  out  and  one  only  persists,  or  both  run  at  point  a. 

However,  the  angle  under  which  the  two  curves,  A  and  C,  inter- 
sect at  a  is  so  small,  that  even  at  a  the  two  arcs  are  not  very 
stable. 
12 


178 


ELECTRIC  CIRCUITS 


Furthermore,  a  small  change  in  either  of  the  two  curves,  A  or  C, 
results  in  the  two  points  of  intersection  a  and  Evanishing.  Thus, 
if  r  is  reduced  from  40  ohms  to  35  ohms,  the  curve  C  changes  to  C", 
shown  dotted  in  Fig.  85,  and  as  the  latter  does  not  intersect  A 
except  at  the  unstable  point  c,  parallel  operation  is  not  possible. 

That  is,  two  such  arcs  can  be  operated  in  parallel  only  over  a 
limited  range  of  conditions,  and  even  then  the  parallel  operation 
is  not  very  stable. 

The  preceding  may  illustrate  the  effect  of  resistance  on  the 
stability  of  operation  of  arcs. 

Similarly,  other  conditions  can  be  investigated,  as  the  stability 

CAPACITY  SHUNTING  ARC 


FIG.  86. 

condition  of  arcs  with  resistance  in  series  and  in  shunt,  on  constant, 
voltage  supply,  etc. 
91.  Let 

e  =  E 

be  the  voltage  consumed  by  a  circuit,  A}  Fig.  86,  when  traversed 
by  a  current 

i  =  I. 
If,  then,  in  this  circuit  the  current  changes  by  57,  to 

i  =  I  +  57, 

the  voltage  consumed  by  the  circuit  changes  by  5  E,  to 

e  =  E  ±  5  E, 

and  the  change  of  voltage  is  of  the  same  sign  as  that  of  the  current 
producing  it,  if  A  is  a  resistance  or  other  circuit  in  which  the 


INSTABILITY  OF  CIRCUITS  179 

voltage  rises  with  the  current,  or  is  of  opposite  sign,  if  the  circuit, 
A,  has  a  dropping  volt-ampere  characteristic,  as  an  arc. 

Suppose  now  the  circuit,  A,  is  shunted  by  a  condenser,  C.  As 
long  as  current,  i,  and  voltage,  e,  in  the  circuit,  A,  are  constant,  no 
current  passes  through  the  condenser,  C.  If,  however,  the  voltage 
of  A  changes,  a  current,  ii,  passes  through  the  condenser,  given  by 
the  equation 

i-cg.  (26) 

If,  then,  the  supply  current,  7,  suddenly  changes  by  67,  from 
7  to  7  +  67,  and  the  circuit,  A,  is  a  dead  resistance,  r,  without  the 
condenser,  C,  the  voltage  of  A  would  just  as  suddenly  change, 
from  E  to  E  +  8E.  By  (26)  this  would,  however,  give  an  infinite 
current,  ii,  in  the  condenser.  However,  the  current  in  the  con- 
denser can  not  exceed  67,  as  with 

ii  =  57 

at  the  moment  of  supply  current  change,  the  total  excess  current 
would  in  the  first  moment  flow  through  the  condenser,  and  the 
circuit,  A,  thus  in  this  moment  not  change  in  current  or  voltage. 
A  finite  current  in  the  condenser,  C,  requires  a  finite  rate  of 
change  of  e  in  the  circuit,  A,  starting  from  the  previous  value,  E, 
at  the  starting  moment,  the  time,  t  =  0. 

Thus,  if  i  =  current,  e  =  voltage  of  circuit,  A,  at  time,  t,  after 
the  increase  of  the  supply  current,  7,  by  67,  it  is 
current  in  condenser, 

de 

tl  =  Cdt' 
current  in  circuit,  A, 

i  =  I  +  67  -  t!  (27) 

thus,  voltage  of  circuit,  A,  of  resistance,  r, 

e  =  ri 
=  rl  +  r67  -  n'i  (28) 

substituting  (26)  into  (28),  gives 

e  =  r(I  +  67)  -  rC  ^ 
or, 

K/+V.-R?*  (29) 

integrated  by 


180  ELECTRIC  CIRCUITS 

(30) 


e  =  rl  -  rdl  (1  -  e 

t 

=  E  -  8E(l  -e~^) 


since  e  =  E  for  t  =  0  is  the  terminal  condition  which  determines 
the  integration  constant. 

With  a  sudden  change  of  the  supply  current,  I,  by  67,  as  shown 
by  the  dotted  lines,  7,  in  Fig.  86,  the  voltage,  e,  and  current,  i,  in 
the  circuit,  A,  and  the  current,  ii,  in  the  condenser,  C,  thus  change 
by  the  exponential  transients  shown  in  Fig.  86  as  e,  i  and  i\. 

92.  Suppose  now,  however,  that  the  circuit,  A,  has  a  dropping 
volt-ampere  characteristic,  is  an  arc. 

A  sudden  decrease  of  the  supply  current,  /  by  67,  to  7  —  57, 

6 
would  by  the  arc  characteristic,  e  =  a  -f  —/=-.,  cause  an  increase  of 


the  voltage  of  circuit,  A,  from  E  to  E  +  8E.  Such  a  sudden 
increase  of  E  would  send  an  infinite  current  through  C,  that  is, 
all  the  supply  current  would  momentarily  go  through  the  con- 
denser, C,  none  through  the  arc,  A,  and  the  latter  would  thus  go 
out,  and  that,  no  matter  how  small  the  condenser  capacity,  C. 
Thus,  with  the  condenser  in  shunt  to  the  circuit,  A,  the  volt  age,  A, 
can  not  vary  instantly,  but  at  a  decrease  of  the  supply  current,  7, 
by  57,  the  voltage  of  A  at  the  first  moment  must  remain  the  same, 
E,  and  the  current  in  A  thus  must  remain  also,  and  as  the  supply 
current  has  decreased  by  67,  the  condenser,  C,  thus  must  feed  the 
current,  67,  back  into  the  arc,  A.  This,  however,  requires  a  de- 
creasing voltage  rating  of  A,  at  decreasing  supply  current,  and 
this  is  not  the  case  with  an  arc. 

Inversely,  a  sudden  increase  of  7,  by  67,  decreases  the  voltage 
of  A,  thus  causes  the  condenser,  C,  to  discharge  into  A,  still  further 
decreases  its  voltage,  and  the  condenser  momentarily  short-cir- 
cuits through  the  arc,  A  ;  but  as  soon  as  it  has  discharged  and  the 
arc  voltage  again  rises  with  the  decreasing  current,  the  condenser, 
C,  robs  the  arc,  A,  and  puts  it  out. 

Thus,  even  a  small  condenser  in  shunt  to  an  arc  makes  it  un- 
stable and  puts  it  out. 

If  a  resistance,  r0,  is  inserted  in  series  to  the  arc  in  the  circuit,  A, 
stability  results  if  the  resistance  is  sufficient  to  give  a  rising  volt- 
ampere  characteristic,  as  discussed  previously. 

Resistance  in  series  to  the  condenser,  C,  also  produces  stability, 
if  sufficiently  large:  with  a  sudden  change  of  voltage  in  the  arc 


INSTABILITY  OF  CIRCUITS 


181 


circuit,  A,  the  condenser  acts  as  a  short-circuit  in  the  first  moment, 
passing  the  current  without  voltage  drop,  and  the  voltage  thus 
has  to  be  taken  up  by  the  shunt  resistance,  r\,  giving  the  same  con- 
dition of  stability  as  with  an  arc  in  a  constant-current  circuit, 
shunted  by  a  resistance,  paragraph  89. 

If,  in  addition  to  the  capacity,  (7,  an  inductance,  L,  and  some  re- 
sistance, r,  are  shunted  across  the  circuit,  A,  of  a  rising  volt-ampere 
characteristic,  as  shown  in  Fig.  87,  the  readjustment  occurring  at 
a  sudden  change  of  the  supply  current,  7,  is  not  exponential,  as  in 
Fig.  86,  but  oscillatory,  as  in  Fig.  87.  As  in  the  circuit,  A,  assum- 
ing it  consists  of  a  resistance,  r,  current  and  voltage  vary 
simultaneously  or  in  phase,  current  and  voltage  in  the  condenser 
branch  circuit  also  must  be  in  phase  with  each  other,  that  is,  the 


FIG.  87. 


frequency  of  the  oscillation  in  Fig.  87  is  that  at  which  capacity,  C, 
and  inductance,  L,  balance,  or  is  the  resonance  frequency. 

If  circuit,  A,  in  Fig.  87  is  an  arc  circuit,  and  the  resistance,  r,  in 
the  shunt  circuit  small,  instability  again  results,  in  the  same  man- 
ner as  discussed  before. 

93.  Another  way  of  looking  at  the  phenomena  resulting  from  a 
condenser,  C,  shunting  a  circuit,  A,  is: 

Suppose  in  Fig.  86  at  constant-supply  current,  /,  the  current  in 
the  circuit,  A,  should  begin  to  decrease,  for  some  reason  or  another. 
Assuming  as  simplest  case,  a  uniform  decrease  of  current. 

The  current  in  the  circuit,  A,  then  can  be  represented  by 


('  -  s) 


(31) 


where  to  is  the  time  which  would  be  required  for  a  uniform  de- 
crease down  to  nothing. 


182  ELECTRIC  CIRCUITS 

At  constant-supply  current,  7,  the  condenser  thus  must  absorb 
the  decrease  of  current  in  A,  that  is,  the  condenser  current  is 

ii-Ij-  (32) 

to 

With  decrease  of  current,  i,  if  A  is  a  circuit  with  rising  character- 
istic, for  instance,  an  ohmic  resistance,  the  voltage  of  A  decreases. 
The  voltage  at  the  condenser  increases  by  the  increasing  charging 
current,  t'i,  thus  the  condenser  voltage  tends  to  rise  over  the  cir- 
cuit voltage  of  A,  and  thus  checks  the  decrease  of  the  voltage  and 
thus  of  the  current  in  A.  Thus,  the  conditions  are  stable. 

Suppose,  however,  A  is  an  arc. 

A  decrease  of  the  current  in  A  then  causes  an  increase  of  the 
voltage  consumed  by  A,  the  arc  voltage,  eQ. 

The  same  decrease  of  the  current  in  A,  by  deflecting  the  current 
into  the  condenser,  causes  an  increase  of  the  voltage  consumed  by 
Cj  the  condenser  voltage,  e\. 

If,  now,  at  a  decrease  of  the  arc  current,  i,  the  arc  voltage,  e0,  rises 
faster  than  the  condenser  voltage,  ei,  the  increase  of  eQ  over  ei  de- 
flects still  more  current  from  A  into  C,  that  is,  the  arc  current 
decreases  and  the  condenser  current  increases  at  increasing  rate, 
until  the  arc  current  has  decreased  to  zero,  that  is,  the  arc  has 
been  put  out.  In  this  case,  the  condenser  thus  produces  in- 
stability of  the  arc. 

If,  however,  eQ  increases  slower  than  ei,  that  is,  the  condenser 
voltage  increases  faster  than  the  arc  voltage,  the  condenser,  C, 
shifts  current  over  into  the  arc  circuit,  A,  that  is,  the  decrease  of 
current  in  the  arc  circuit  checks  itself,  and  the  condition  becomes 
stable. 

The  voltage  rise  at  the  condenser  is  given  by 

d?  =  I  •  . 

hence,  by  (32), 

de  _    II 
dt  ~  t0C 

from  the  volt-ampere  characteristic  of  the  arc, 

e  =  a  +  -4=  (34) 

follows, 


INSTABILITY  OF  CIRCUITS  183 


the  voltage  rise  at  the  arc  terminals, 

de  _  b       di_ 

dt  2i-\/l,  dt 

and,  by  (31), 

di         _  /. 
dt  ~         to' 

hence,  substituted  into  (34), 

de          '&/ 


(35) 


(36) 


dt       2  tQi\/i 

The  condition  of  stability  is,  that  the  voltage  rise  at  the  con- 
denser, (33),  is  greater  than  that  at  the  arc,  (36),  thus, 

tl  .         bl 


t0C  "      2  tglT/i 

or, 

*"/7>l  (37) 


bC 

or,  substituting  for  t  from  equation  (31),  gives 
2 


(38) 


bC  *  (39) 

thus  is  the  stability  limit. 

94.  Integrating  (33)  and  substituting  the  terminal  condition: 
t  =  0;  e  =  E,  gives 


as  the  condition  of  stability,  and 

2  to'vT  (/  -  i) 


as  the  equation  of  the  voltage  at  the  condenser  terminals. 
Substitute  (31)  into  (34)  gives 


as  the  equation  of  the  arc  voltage. 
For, 

a  =  35, 
6  =  200, 
/  =  3, 

hence, 


184 


and 


ELECTRIC  CIRCUITS 
E  =  151, 


t0  =  10~4  sec., 
and,  for  the  three  values  of  capacity, 

C  =  10-6 
0.75  X  10-6 
0.5  X  10-6 


CAPACITY  SHUNTING  ARC 


FIG.  88. 

the  curves  of  the  arc  voltage,  e0, 

and  of  the  condenser  voltage,  d,  e2}  e3, 

are  shown  on  Fig.  88, 

together  with  the  values  of  i  and  i\. 

As  seen,  e\  is  below  eQ  over  the  entire  range.  That  is,  1  mf. 
makes  the  arc  unstable  over  the  entire  range.  0.5  mf.,  e3,  gives 
instability  up  to  about  t  =  0.25  X  10~4  sec.,  then  stability  results. 
With  0.75  mf.,  e2,  there  is  a  narrow  range  of  stability,  between 


INSTABILITY  OF  CIRCUITS  185 

and  7J4  X  10~4  sec.,  before  and  after  this  instability 
exists. 

From  equation  (37),  the  condition  of  stability,  it  follows  that 
for  small  values  of  t,  that  is,  small  current  fluctuations,  the  con- 
ditions are  always  unstable.  That  is,  no  matter  how  small  a 
condenser  is,  it  always  has  an  effect  in  increasing  the  current 
fluctuations  in  the  arc,  the  more  so,  the  higher  the  capacity,  until 
conditions  become  entirely  unstable. 

From  equations  (40)  and  (41)  follows  as  the  stability  limit 

b  t2! 

&  H T^ — i —         =  E  +  0  A  ^t 


or,  expanded  into  a  series, 

"  V7  l          2TQ 
cancelling  E  =  a  -f-  ~~7J  and  rearranging,  gives 

(42) 


thus,  at  the  time, 

bC 

','/•      :\       tl  =  ^'  .  .       ;   -    - 

the  condition  changes  from  unstable  to  stable. 

As  ti  must  be  smaller  than  tQ,  the  total  time  of  change,  it 
follows : 

(43) 


or, 

C  <  t-°i-p  (44) 

are  expressions  of  the  (approximate)  stability  limit  of  an  arc  with 
condenser  shunt. 
As  seen  from  (44), 

the  larger  t0  is,  that  is,  the  slower  the  arc  changes,  the  larger  is 
the  permissible  shunted  capacity,  and  inversely. 
As  an  instance,  let 

b  =  200, 
/=      3, 
and 


186 


ELECTRIC  CIRCUITS 


(a)     tQ  =  10~3, 

which  is  probably  the  approximate  magnitude  in  the  carbon  arc. 
This  gives 

C  <  26  mf. 
Let: 

(6)  tQ  =  10-5, 

which  is  probably  the  approximate  magnitude  in  the  mercury  arc. 
This  gives 

C  <  0.26  mf. 

95.  Consider  the  case  of  a  circuit,  A,  Fig.  87,  supplied  by  a 
constant  current,  /,  but  shunted  by  a  capacity,  C,  inductance,  L, 
and  resistance,  r,  in  series. 


RESONATING  CIRCUIT 
SHUNTING  ARC 


FIG.  89. 

As  long  as  the  current  in  the  circuit,  A — whether  resistance  or 
arc — is  steady,  no  current  passes  the  condenser  circuit,  and  the 
current  and  voltage  in  A  thus  are  constant,  i  =  I,  e  =  eQ. 

Suppose  now  a  pulsation  of  the  current,  i,  should  be  produced 
in  circuit,  A,  as  shown  as  i  in  Fig.  89.  Then,  with  constant-sup- 
ply current,  7,  an  alternating  current, 

i\  =  7  -  i, 

would  traverse  the  condenser  circuit,  C,  since  the  continuous  com- 
ponent of  current  can  not  traverse  the  condenser,  C. 


INSTABILITY  OF  CIRCUITS  187 

Due  to  the  pulsation  of  current,  i  in  A,  the  voltage,  e,  of  cir- 
cuit, A,  would  pulsate  also.  These  voltage  pulsations  are  in  the 
same  direction  as  the  current  pulsation,  if  A  is  a  resistance,  in 
opposite  direction,  if  A  is  an  arc;  in  either  case,  however,  they 
are  in  phase  with  the  current  pulsation,  and  the  alternating  vol- 
tage on  the  condenser, 

61  =  eo  —  e, 

thus  is  in  phase  with  the  alternating  current,  ii,  that  is,  capacity, 
C,  and  inductance,  L,  neutralize. 

Thus,  the  only  pulsation  of  current  and  voltage,  which  could 
occur  in  a  circuit,  A,  shunted  by  capacity  and  inductance,  is  that 
of  the  resonance  frequency  of  capacity  and  inductance. 

Suppose  the  circuit,  A ,  is  a  dead  resistance.  The  voltage  pulsa- 
tion produced  by  a  current  pulsation,  i,  in  this  circuit  then  would 
be  in  the  same  direction  as  i,  that  is,  would  be  as  shown  in  dotted 
line  by  e'  in  Fig.  89.  In  the  condenser  circuit,  C,  the  alternat- 
ing component  of  voltage  thus  would  be 

e'i  =  e'  -  e0, 

thus  would  be  in  opposition  to  the  alternating  current,  tj,  as  shown 
in  Fig.  89  in  dotted  line.  That  is,  it  would  require  a  supply  of 
power  to  maintain  such  pulsation. 

Thus,  with  a  dead  resistance  as  circuit,  A,  or  in  general  with  A 
as  a  circuit  of  rising  volt-ampere  characteristic,  the  maintenance 
of  a  resonance  pulsation  of  current  and  voltage  between  A  and  C, 
at  constant  current,  7,  requires  a  supply  of  alternating-current 
power  in  the  condenser  circuit,  and  without  such  power  supply 
the  pulsation  could  not  exist,  hence,  if  started,  would  rapidly  die 
out,  as  oscillation,  as  shown  in  Fig.  87. 

96.  Suppose,  however,  A  is  an  arc.  A  current  pulsation,  it  then 
gives  a  voltage  pulsation  in  opposite  direction,  as  shown  by  e 
in  Fig.  89,  and  the  alternating  current,  i\  —  I  —  i,  and  the  alter- 
nating voltage,  ei  =  e  —  e0,  in  the  condenser  circuit,  thus  would 
be  in  phase  with  each  other,  as  shown  by  ii  and  e\  in  Fig.  89. 
That  is,  they  would  represent  power  generation,  or  rather  trans- 
formation of  power  from  the  constant  direct-current  supply,  7, 
into  the  alternating-current  resonating  condenser  circuit,  C. 

Thus,  such  a  local  pulsation  of  the  arc  current,  i,  and  corre- 
sponding alternating  current,  ii,  in  the  condenser  circuit,  if  once 
started,  would  maintain  itself  without  external  power  supply, 


188  ELECTRIC  CIRCUITS 

and  would  even  be  able  to  supply  the  power  represented  by  vol- 
tage, ei,  with  current,  ii,  into  an  external  circuit,  as  the  resistance, 
r,  shown  in  Fig.  87,  or  through  a  transformer  into  a  wireless  send- 
ing circuit,  etc. 

Thus,  due  to  the  dropping  arc  characteristic,  an  arc  shunted 
by  capacity  and  inductance,  on  a  constant-current  supply,  be- 
comes a  generator  of  alternating-current  power,  of  the  frequency 
set  by  the  resonance  of  C  and  L. 

If  the  resistance,  r,  or  in  general,  the  load  on  the  oscillating  cir- 
cuit, C,  is  greater  than  r*i  =  — ,  that  is,  if  a  higher  voltage  would  be 

1*1 

required  to  send  the  current,  ii,  through  the  resistance,  r,  than  the 
voltage,  eij  generated  by  the  oscillating  arc,  A,  the  pulsations  die 
out  as  oscillations. 

If  r  is  less  than  -^,  the  pulsations  increase  in  amplitude,  that  is, 
current,  ii,  and  voltage,  <?i,  increase,  until  either,  by  the  internal 
reaction  in  the  arc,  the  ratio,  -^,  drops  to  equality  with  the  effective 
resistance  of  the  load,  r,  and  stability  of  oscillation  is  reached, 
or,  if  —  never  falls  to  equality  with  r — for  instance,  if  r  =  0,  the 

oscillations  increase  up  to  the  destruction  of  the  circuit:  the 
extinction  of  the  arc. 

If,  in  the  latter  case,  the  voltage  back  of  the  supply  current,  I, 
is  sufficiently  high  to  restart  the  arc,  A,  the  phenomena  repeats, 
and  we  have  a  series  of  successive  arc  oscillations,  each  rising  until 
it  puts  the  arc  out,  and  then  the  arc  restarts. 

We  thus  have  here  the  mechanism  which  produces  a  cumulative 
oscillation,  that  is,  a  transient,  which  does  not  die  out,  but  in- 
creases in  amplitude,  until  the  increasing  energy  losses  limit  its 
further  increase,  or  until  it  destroys  the  circuit,  and  in  the 
latter  case,  it  may  become  recurrent. 

It  is  very  important  to  realize  in  electrical  engineering,  that 

any  electric  circuit  with  dropping  volt-ampere  characteristic  is 

.  capable  of  transforming  power  into  a  cumulative  oscillation,  and 

thereby  is  able  under  favorable  conditions  to  produce  cumulative 

oscillations,  such  as  hunting,  etc. 

Where  the  arc  oscillations  limit  themselves,  and  the  alternating 
current  and  voltage  in  the  condenser  circuit  thus  reach  a  constant 
value,  the  arc  often  is  called  a  "singing  arc,"  due  to  the  musical 
note  given  by  the  alternating  wave.  Where  the  arc  oscillations 


INSTABILITY  OF  CIRCUITS 


189 


rise  cumulatively  to  interruption,  and  the  arc  then  restarts  by 
the  supply  voltage  and  repeats  the  same  phenomenon,  it  may 
be  called  a  "rasping  arc,"  by  the  harsh  noise  produced  by  the 
interrupted  cumulative  oscillation. 


4.0     4.5     5.0     6.5     C.O     65     70     75 


FIG.  94. 

Figs.  90  and  91  give  oscillograms  of  singing  arcs;  Figs.  92  and 
93,  of  rasping  arcs,  90  and  92  in  circuits  with  massed  constants, 
91  and  93  in  transmission  lines. 

97.  As  an  illustration,  let  curve,  A, in  Fig.  94  represent  the  volt- 
ampere  characteristic  of  an  arc,  and  assume  that  this  arc  is  operat- 
ing steadily  at  /  amp.,  consuming  eQ  volts. 


190  ELECTRIC  CIRCUITS 

Suppose  this  arc  is  shunted  by  capacity,  C,  inductance,  L,  and 
resistance,  r,  as  shown  in  Fig.  87. 

For  a  small  pulsation  of  the  arc  current  around  its  average 
value  i  =  I,  the  corresponding  voltage  pulsation  is  given  by 

de  b 


Or,  in  general,  for  any  pulsation  of  current,  i,  by  di,  between 
ir  and  i" ',  around  the  mean  value  7,  the  corresponding  voltage 
pulsation  de,  between  ef  and  e",  is  given  by  the  volt-ampere 
characteristic  of  the  arc,  A,  as 

de  _     _  e"  -  e' 
di  i"  -  i'' 

e\  =  8e  thus  is  the  voltage,  made  available  for  the  condenser 
circuit,  by  the  arc  pulsation,  and  in  phase  with  the  current, 
ii  =  —  di  in  the  condenser  circuit,  and 

8e  _  e-  _  e/ 
di      i"  -  i" 

thus  is  the  permissible  effective  resistance  in  the  condenser 
circuit,  that  is,  the  maximum  value  of  resistance,  through  which 
the  pulsating  arc  can  maintain  its  alternating  power  supply: 
with  a  larger  resistance,  the  oscillations  die  out;  with  a  smaller 
resistance,  they  increase. 

From  the  arc  characteristic,  A,  thus  can  be  derived  a  curve 

of  effective  resistances,  R}  as  the  values  of  — .,  for   pulsations 

ul 

between  i  +  di  and  i  —  di,  and  such  a  curve  is  shown  as  R  in 
Fig.  94. 

We  may  say,  that  the  arc,  when  shunted  by  an  oscillating 
circuit,  has  an  effective  negative  resistance, 


and  thereby  generates  alternating  power,  from  the  consumed 
direct-current  power,  and  is  able  to  supply  alternating  power 
through  an  effective  resistance  of  the  oscillating  circuit,  of 

ff  &e 

R        si' 


INSTABILITY  OF  CIRCUITS  191 

The  arc  characteristic  in  Fig.  94  is  drawn  with  the  equation 


V* 
and  for 

i  =  I  =  3  amp.  as  mean  value, 
the  values  of  the  effective  resistance,  R,  increase  from 

R  =  -  ^  =  18.5  ohms 
di 

for  very  small  oscillations,  to 

R  =  20.3  ohms 
for  oscillations  of  1  amp.,  between  i  =  2  and  i  =  4,  to 

R  =  27.5  ohms 

for  oscillations  of  2  amp.,  between  i  =  1  and  i   =   5,  etc. 

Thus,  if  with  this  oscillating  arc,  Figs.  87  and  94,  a  load  resist- 
ance r  <  18.5  ohms  is  used,  oscillation  starts  immediately,  and 
cumulatively  increases. 

If  the  resistance,  r,  is  greater  than  18.5  ohms,  for  instance,  is 

r  =  22.5  ohms, 

then  no  oscillation  starts  spontaneously,  but  the  arc  runs  steady, 
and  no  appreciable  current  passes  through  the  condenser  circuit. 
But  if  once  the  current  in  the  arc  is  brought  below  1.5  amp.,  or 
above  4.5  amp.,  the  oscillation  begins  and  cumulatively  increases, 
since  for  oscillations  of  an  amplitude  greater  than  between  1.5  and 
4.5  amp.,  the  effective  resistance,  R,  is  greater  than  22.5  ohms. 

In  either  case,  however,  as  soon  as  an  oscillation  starts,  it  cumu- 
latively increases,  since  the  effective  resistance,  R,  steadily  in- 
creases with  increase  of  the  amplitude  of  oscillation.  That  is, 
stability  of  oscillation,  or  a  "singing  arc"  can  not  be  reached,  but 
an  oscillation,  once  started,  proceeds  to  the  extinction  of  the  arc, 
and  only  a  "  rasping  arc"  could  be  produced. 

98.  However,  the  arc  characteristic,  A,  of  Fig.  94  is  the  sta- 
tionary characteristic,  that  is,  the  volt-ampere  relation  at  constant 
current,  i,  and  voltage,  e. 

If  current,  i,  and  thus  voltage,  e,  rapidly  fluctuate,  the  arc  char- 
acteristic, A,  changes,  and  more  or  less  flattens  out.  That  is,  for 


192 


ELECTRIC  CIRCUITS 


any  value  of  the  current,  i,  the  volume  of  the  arc  stream  and  the 
temperature  of  the  arc  terminals,  still  partly  correspond  to  pre- 
vious values  of  current,  thus  are  lower  for  rising,  higher  for 
decreasing  current,  and  as  the  result,  the  arc  voltage,  e,  which  de- 


TRANSIENT 

VOLT-AMPERE 

CHARACTERISTIC 

OF  ARCS 


1.0     1.5     2.0     2.5     3.0     35     4.0 


FIG.  95. 


pends  on  the  resistance  of  the  arc  stream  and  the  potential  drop 
of  the  terminals,  is  different,  the  variation  of  voltage,  for  the  same 
variation  of  current,  is  less,  and  the  effective  negative  arc  resist- 
ance thereby  is  lowered,  or  may  entirely  vanish. 

Fig.  95  shows  a  number  of  such  transient  arc  characteristics, 


INSTABILITY  OF  CIRCUITS  193 

estimated  from  oscillographic  tests  of  alternating  arcs,  and  their 
corresponding  effective  resistances,  R. 

They  are: 

(A)  Carbon. 

(#)  Hard  carbon. 

(C)  Acheson  graphite. 

(D)  Titanium  carbid. 

(E)  Hard  carbon,  stationary  characteristic. 

(F)  Titanium  carbid,  stationary  characteristic. 

As  seen  from  the  curves  of  R  in  the  upper  part  of  Fig.  95,  the 
effective  resistances,  R,  which  represent  the  alternating  power 
generated  by  the  oscillating  arc,  are  much  lower  with  the  transi- 
ent arc  characteristic,  than  would  be  with  the  permanent  arc 
characteristic  in  Fig.  94. 

Curve  D,  titanium  carbide,  gives  under  these  conditions  an 
unstable  or  "rasping"  arc.  That  is,  with  a  resistance  in  the  con- 
denser circuit  of  less  than  R  =  3.8  ohms,  the  oscillation  starts 
spontaneously  and  cumulatively  increases  to  the  extinction  of  the 
arc;  with  a  resistance  of  more  than  3.8  ohms,  the  oscillation  does 
not  spontaneously  start,  but  if  once  started  with  an  amplitude 
which  brings  the  value  of  R  from  curve,!),  above  that  of  the  resist- 
ance in  the  condenser  circuit,  cumulative  oscillation  occurs. 

With  the  carbon  arc,  A,  no  oscillations  can  occur  under  any 
condition,  the  effective  resistance,^,  is  negative,  and  the  arc  char- 
acteristic rising. 

With  the  hard  carbon  arc,  B,  an  oscillation  starts  with  a  resist- 
ance less  than  2.4  ohms,  cumulatively  increases,  but  its  amplitude 
finally  limits  itself,  to  1.45  amp.  if  the  resistance  in  the  oscillating 
circuit  is  zero,  to  1.05  amp.  with  2  ohms  resistance,  etc.,  as  seen 
from  the  curve,  B,  in  the  upper  part  of  Fig.  95.  Even  with  more 
than  2.4  ohms  resistance,  up  to  2.6  ohms  resistance,  an  oscillation 
can  exist,  if  once  started,  as  the  curve  of  R,  starting  from  R  =  2.4 
ohms  at  ii  =  0,  rises  to  R  =  2.6  ohms  at  i\  =  0.75,  and  then 
drops  to  zero  at  i\  =  1.45  ohms,  and  beyond  this  becomes 
negative. 

The  curve,  C,  of  Acheson  graphite,  starts  with  a  resistance  R  = 
10.8  ohms,  but  the  resistance,  R,  steadily  drops  with  increasing 
oscillating  current,  it,  down  to  zero  at  i\  =  2.4  amp.  Thus,  with 
a  resistance  in  the  condenser  circuit,  of  10  ohms,  the  oscillations 
would  have  an  amplitude  of  ii  =  0.9  amp. ;  with  8  ohms  resistance 
an  amplitude  of  2.1  amp.,  etc. 


194  ELECTRIC  CIRCUITS 

From  these  curves  of  R,  Fig.  95,  the  regulation  curves  of  the 
alternating-current  generation  could  now  be  constructed. 

It  is  interesting  to  note,  that  in  many  of  these  transient  arc 
characteristics,  Fig.  95,  the  voltage  does  not  indefinitely  rise  with 
decreasing  current,  but  reaches  a  maximum  and  then  decreases 
again,  in  B  and  C,  and  the  oscillation  resistance,  that  is,  the  re- 
sistance through  which  an  alternating  current  can  be  maintained 
by  the  oscillating  arc,  thus  decreases  with  increasing  amplitude  of 
the  oscillation.  Thus,  if  the  resistance  in  the  oscillating  con- 
denser circuit  is  less  than  the  permissible  maximum,  an  oscilla- 
tion starts,  cumulatively  increases,  but  finally  limits  itself  in 
amplitude. 

The  decrease  of  the  arc  voltage  with  decreasing  current,  for  low 
values  of  current  in  a  rapidly  fluctuating  arc,  is  due  to  the  time 
lag  of  the  arc  voltage  behind  the  current. 

99.  The  arc  voltage,  e,  consists  of  the  arc  terminal  drop,  a,  and 
the  arc  stream  voltage,  e\\ 

e  —  a  +  e\. 

The  stream  voltage,  e\,  is  the  voltage  consumed  in  the  effective 
resistance  of  the  arc  stream;  but  as  the  arc  stream  is  produced 
by  the  current,  the  volume  of  the  arc  stream  and  its  resistance 
thus  depends  on  the  current,  i,  in  the  arc,  that  is,  the  stream  vol- 
tage is 

b 


and  the  resistance  of  the  arc  stream  thus 

6x  b 

ri  =:  i  ==  Svt 

Thus,  if, 

a  =  35 
b  =  200, 
for 

i    =  2  amp.,  it  is 
ri  =  70.7  ohms, 
61  =  141.4  volts, 
e    =  176.4  volts. 

But,  if  the  arc  current  rapidly  varies,  for  instance  decreases, 
then,  when  the  current  in  the  arc  is  ii,  the  volume  of  the  arc  stream 


INSTABILITY  OF  CIRCUITS  195 

and  thus  its  resistance  is  still  that  corresponding  to  the  previous 
current,  i'\. 

If  thus,  at  the  moment  where  the  current  in  the  arc  has  become 

i\  =  2  amp., 

the  arc  stream  still  has  the  volume  and  thus  the  resistance 
corresponding  to  the  previous  current, 

i'i  =  3  amp., 
this  resistance  is 

200 
r'i  =  — -7=  =  38. 5  ohms, 

and  the  stream  voltage,  at  the  current 

i\  =  2  amp., 

but  with  the  stream  resistance,  r'i,  corresponding  to  the  previous 
current,  i'\  =  3  amp.,  thus  is 

e'i  =  r\ii 
=  77  volts, 

instead  of  e\  =  141.4  volts,  as  it  would  be  under  stationary 
conditions. 

That  is,  the  stream  voltage  and  thus  the  total  arc  voltage  at 
rapidly  decreasing  current  is  lower,  at  rapidly  increasing  current 
higher  than  at  stationary  current. 

With  a  periodically  pulsating  current,  it  follows  herefrom,  that 
at  the  extreme  values  of  current — maximum  and  minimum —  the 
voltage  has  not  yet  reached  the  extreme  values  corresponding 
to  these  currents,  that  is,  the  amplitude  of  voltage  pulsation  is 
reduced.  This  means  the  transient  volt-ampere  characteristic 
of  the  arc  is  flattened  out,  compared  with  the  permanent  charac- 
teristic, and  caused  to  bend  downward  at  low  currents,  as  shown 
by  C  and  B  in  Fig.  95. 

Assuming  a  sinusoidal  pulsation  of  the  current  in  the  arc  and 
assuming  the  arc  stream  resistance  to  lag  behind  the  current  by  a 
suitable  distance,  we  then  get,  from  the  stationary  volt-ampere 
characteristic  of  the  arc,  the  transient  characteristics. 

Thus  in  Fig.  96,  from  the  stationary  arc  characteristic,  S,  the 
transient  arc  characteristic,  T,  is  derived.  In  this  figure  is  shown 
as  S  and  T  the  effective  resistance  corresponding  to  the  stationary 
characteristic,  S,  respectively  the  transient  characteristic,  T. 


196 


ELECTRIC  CIRCUITS 


As  seen,  the  stationary  characteristic,  S,  gives  an  arc  oscillation 
which  is  cumulative  and  self-destructive,  that  is,  the  effective 
resistance,  R,  rises  indefinitely  with  increasing  amplitude  of 
pulsation.  The  transient  characteristic,  however,  gives  an  effect- 
ive resistance,  R,  which  with  increasing  amplitude  of  pulsation 


\ 

v: 

\ 

OSCILLATION  RESISTANCE 

200 

Ks 

OF  ARC 

190 

^ 

Ns 

\ 

_180 

T 

V 

\ 

-170 

S 

IfiO 

\ 

-J150 

S 

^ 

^ 



_140 
_130 

T 

jh 

^ 

120 

r 

40 

— 

— 

i 







/ 

35 

S 

/ 

30 

/ 

/ 

25 

- 

_^ 

^ 

/ 

20 

_—  —  • 

T 

15 

"^ 

N 

10 

-? 

5-2 

.0-1 

5-1 

0  -.. 

j      , 

+  . 

5+1 

0  +  1 

5-1-2 

0  +  2 

J 

=  i, 

. 

i       110      1 

5     2 

0     2 

5     3 

0     315     410    4J5     5]0     5 

5     6 

o    •• 

I 

FIG.  96. 

first  increases,  but  then  decreases  again,  down  to  zero,  so  that  the 
cumulative  oscillations  produced  by  this  arc  are  self-limiting, 
increase  in  amplitude  only  up  to  the  value,  where  the  effective 
resistance,  R,  has  fallen  to  the  value  corresponding  to  the  load  on 
the  oscillating  circuit. 


INSTABILITY  OF  CIRCUITS 


197 


As  further  illustration,  from  the  stationary  volt-ampere  char- 
acteristic of  the  titanium  arc,  shown  as  F  in  Fig.  95,  values  of  the 
transient  characteristic  have  been  calculated  and  are  shown  in 
Fig.  95  by  crosses.  As  seen,  they  fairly  well  coincide  with  the 
transient  volt-ampere  characteristic,  Z>,  of  the  titanium  arc,  at 
least  for  the  larger  currents. 


FIG.  97. 

In  the  electric  arc  we  thus  have  an  electric  circuit  with  dropping 
volt-ampere  characteristic.  Such  a  circuit  is  unstable  under 
various  conditions  which  may  occur  in  industrial  circuits,  and 
thereby  may  be,  and  frequently  is,  the  source  of  instability  of 
electric  circuits,  and  of  cumulative  oscillations  appearing  in  such 
circuits. 


198  ELECTRIC  CIRCUITS 

100.  For  instance,  let,  in  Fig.  97,  A  and  B  be  two  conductors 
of  an  ungrounded  high-potential  transmission  line,  and  2  e  the 
voltage  impressed  between  these  two  conductors.  Let  C  repre- 
sent the  ground. 

The  capacity  of  the  conductors,  A  and  5,  against  ground,  then, 
may  be  represented  diagrammatically  by  two  condensers,  C\  and 
£2,  and  the  voltages  from  the  lines  to  ground  by  e\  and  e2.  In  gen- 
eral, the  two  line  capacities  are  equal,  C\  =  C2,  and  the  two  volt- 
ages to  ground  thus  equal  also,  ei  =  e2  =  e,  with  a  single-phase; 

=  —7=  with  a  three-phase  line. 

Assume  now  that  a  ground,  P,  is  brought  near  one  of  the  lines, 
A,  to  within  the  striking  distance  of  the  voltage,  e.  A  discharge 
then  occurs  over  the  conductor,  P.  Such  may  occur  by  the  punc- 
ture of  a  line  insulator  as  not  infrequently  the  case.  Let  r  =  re- 
sistance of  discharge  path,  P.  While  without  this  discharge  path, 
the  voltage  between  A  and  C  would  be  e\  —  e  (assuming  single- 
phase  circuit)  with  a  grounded  conductor,  P,  approaching  line  A 
within  striking  distance  of  voltage,  e,  a  discharge  occurs  over  P 
forming  an  arc,  and  the  circuit  of  the  impressed  voltage,  2  6,  now 
comprises  the  condenser,  Cz,  in  series  to  the  multiple  circuit  of  con- 
denser, Ci,  and  arc,  P,  and  the  condenser,  Ci,  rapidly  discharges, 
voltage,  61,  decreases,  and  the  voltage,  62,  increases.  With  a  de- 
crease of  voltage,  ei,  the  discharge  current,  i,  also  decreases,  and 
the  voltage  consumed  by  the  discharge  arc,  e',  increases  until  the 
two  voltages,  e\  and  e',  cross,  as  shown  in  the  curve  diagram  of 
Fig.  97.  At  this  moment  the  current,  i,  in  the  arc  vanishes,  the 
arc  ceases,  and  the  shunt  of  the  condenser,  Ci,  formed  by  the  dis- 
charge over  P  thus  ceases.  The  voltage,  e\,  then  rises,  e2  decreases 
and  the  two  voltages  tend  toward  equality,  e\  =  e2  =  e.  Before 
this  point  is  reached,  however,  the  voltage,  e\t  has  passed  the  dis- 
ruptive strength  of  the  discharge  gap,  P,  the  discharge  by  the  arc 
over  P  again  starts,  and  the  cycle  thus  repeats  indefinitely. 

In  Fig.  97  are  diagrammatically  sketched  voltage,  ei,  of  con- 
denser, Ci,  the  voltage,  e',  consumed  by  the  discharge  arc  overP, 
and  the  current,  i,  of  this  arc,  under  the  assumption  that  r  is  suffi- 
ciently high  to  make  the  discharge  non-oscillatory.  If  r  is  small, 
each  of  these  successive  discharges  is  an  oscillation. 

Such  an  unstable  circuit  gives  a  continuous  series  of  successive 
discharges,  which  are  single  impulses,  as  in  Fig.  97,  or  more  com- 
monly are  oscillations. 


INSTABILITY  OF  CIRCUITS  199 

If  the  line  conductors,  A  and  B,  in  Fig.  97  have  appreciable  in- 
ductance, as  is  the  case  with  transmission  lines,  in  the  charge  of 
the  condenser,  Ci,  after  it  has  been  discharged  by  the  arc  over  P, 
the  voltage,  ei,  would  rise  beyond  e,  approaching  2  e,  and  the  dis- 
charge would  thus  start  over  P,  even  if  the  disruptive  strength  of 
this  gap  is  higher  than  e,  provided  that  it  is  still  below  the  voltage 
momentarily  reached  by  the  oscillatory  charge  of  the  line  conden- 
ser, PL 

This  combination  of  two  transmission  line  conductors  and  the 
ground  conductor,  P,  approaching  near  line,  A,  to  a  distance  giving 
a  striking  voltage  above  e,  but  below  the  momentary  charging 
voltage,  of  Ci,  then  constitutes  a  circuit  which  has  two  permanent 
conditions,  one  of  stability  and  one  of  instability.  If  the  voltage 
is  gradually  applied,  e\  =  ez  =  e,  the  condition  is  stable,  as  no 
discharge  occurs  over  P.  If,  however,  by  some  means,  as  a  mo- 
mentarily overvoltage,  a  discharge  is  once  produced  over  the 
spark-gap,  P,  the  unstable  condition  of  the  circuit  persists  in  the 
form  of  successive  and  recurrent  discharges. 

101.  Usually,  the  resistance,  r,  of  the  discharge  path  is,  or  after 
a  number  of  recurrent  discharges,  becomes  sufficiently  low  to 
make  the  discharge  oscillatory,  and  a  series  of  recurrent  oscilla- 
tions then  result,  a  so-called  "arcing  ground."  Oscillograms  of 
such  an  arcing  grounds  on  a  30-mile  30-kv.  transmission  line  are 
shown  in  Figs.  98,  99  and  100. 

If,  however,  the  resistance  of  the  discharge  path  is  very  low,  a 
sustained  or  cumulative  oscillation  results,  as  discussed  in  the  pre- 
ceding, that  is,  the  arcing  ground  becomes  a  stationary  oscillation 
of  constant-resonance  frequency,  increasing  cumulatively  in  cur- 
rent and  voltage  amplitude  until  limited  by  increasing  losses  or  by 
destruction  of  apparatus. 

In  transmission  lines,  usually  the  resistance  is  too  high  to  pro- 
duce a  cumulative  oscillation ;  in  underground  cables,  usually  the 
inductance  is  too  low  and  thus  no  cumulative  oscillation  results, 
except  perhaps  sometimes  in  single-conductor  cables,  etc.  In 
the  high-potential  windings  of  large  high-voltage  power  trans- 
formers, however,  as  circuits  of  distributed  capacity,  inductance 
and  resistance,  the  resistance  commonly  is  below  the  value  through 
which  a  cumulative  oscillation  can  be  produced  and  maintained, 
and  in  high-potential  transformers,  destruction  by  high  voltages 
resulting  from  the  cumulative  oscillation  of  some  arc  in  the 


200  ELECTRIC  CIRCUITS 

system,  and  building  up  to  high  stationary  waves,  have  frequently 
been  observed. 

The  " arcing  ground"  as  recurrent  single  impulses,  the  "arcing 
ground  oscillation"  as  more  or  less  rapidly  damped  recurrent 
oscillations  in  transmission  lines — of  frequencies  from  a  few  hun- 
dred to  a  few  thousand  cycles — and  the  "stationary  oscillations" 
causing  destruction  in  high-potential  transformer  windings,  at 
frequencies  of  10,000  to  100,000  cycles,  thus  are  the  same  phenom- 
ena of  the  dropping  arc  characteristic,  causing  permanent  in- 
stability of  the  electric  circuit,  and  differ  from  each  other  merely 
by  the  relative  amount  of  resistance  in  the  discharge  path. 


CHAPTER  XI 

INSTABILITY    OF    CIRCUITS:    INDUCTION    AND    SYN- 
CHRONOUS   MOTORS 

C.     Instability  of  Induction  Motors 

102.  Instability  of  electric  circuits  may  result  from  causes  which 
are  not  electrical:  thus,  mechanical  relations  between  the  torque 
given  by  a  motor  and  the  torque  required  by  its  load,  may  lead  to 
instability. 

Let 

D  =  torque  given  by  a  motor  at  speed,  S,  and 

Df  =  torque  required  by  the  load  at  speed,  S. 

The  motor,  then,  could  theoretically  operate,  that  is,  run  at 
constant  speed,  at  that  speed,  S,  where 

D  =  Dr  (1) 

However,  at  this  speed  and  load,  the  operation  may  be  stable, 
that  is,  the  motor  continue  to  run  indefinitely  at  constant  speed, 
or  the  condition  may  be  unstable,  that  is,  the  speed  change  with 
increasing  rapidity,  until  stability  is  reached  at  some  other  speed, 
or  the  motor  comes  to  a  standstill,  or  it  destroys  itself. 

In  general,  the  motor  torque,  D.  and  the  load  torque,  D', 
change  with  the  speed,  S. 

If,  then, 

dV       dD 
dS>  ~dS 

the  conditions  are  stable,  that  is,  any  change  of  speed,  S,  changes 
the  motor  torque  less  than  the  load  torque,  and  inversely,  and 
thus  checks  itself. 
If,  however, 

dD'       dD 

~dS<dS 

the  operation  is  unstable,  as  a  change  of  speed,  S,  changes  the 
motor  torque,  D,  more  than  the  load  torque,  D',  and  thereby  fur- 
ther increases  the  change  of  speed,  etc. 

dD'_dD  ,.. 

~dS~dS 

201 


202 


ELECTRIC  CIRCUITS 


thus  is  the  expression  of  the  stability  limit. 

For  instance,  assuming  a  load  requiring  a  constant  torque  at  all 
speeds.  The  load  torque  thus  is  given  by  a  horizontal  line 

D'  —  const.  (5) 

in  Fig.  101. 

Let  then  the  speed-torque  curve  of  the  motor  be  represented  by 
the  curve,  D,  in  Fig.  101.  D  approximately  represents  the  torque 
curve  of  a  series  motor.  At  the  constant-load  torque,  D',  the 
motor  runs  at  the  speed,  S  =  0.6,  point  a  of  Fig.  101,  and  the  speed 
is  stable,  as  any  tendency  to  change  of  speed,  checks  itself.  If 

<*  2 


^""^ 

X 

14 

\ 

\ 

D 

10 

\ 

11 

\ 

ai 

10 

u; 

\ 

q 

\ 

g 

\ 

7 

N 

D' 

x 

x, 

a 

fi 

^ 

X 

4 

DO 

^"^ 

*-^ 

•^^ 

a0 

3 

•^^^ 

-  — 

-  — 

1 

.: 

i 

.; 

i 

.c 

.* 

l 

.1 

r 

i 

i 

0 

FIG.  101. 

the  load  torque  decreases  to  D'o,  the  speed  rises  to  S  =  0.865, 
point  a0;  if  the  load  torque  increases  to  D'i,  the  speed  drops  to 
S  =  0.29,  point  «i,  but  the  conditions  are  always  stable,  until 
finally  with  increasing  load  torque,  D',  and  decreasing  speed, 
standstill  is  reached  at  point  a2. 

Let  now  the  speed-torque  curve  of  a  motor  be  represented  by 
D  in  Fig.  102:  the  curve  of  a  squirrel-cage  induction  motor  with 
moderately  high  resistance  secondary.  The  horizontal  line,  D' ', 
corresponding  to  a  load  torque  of  D'  =  10,  intersects  D  at  two 
points,  a  and  b. 


INSTABILITY  OF  CIRCUITS 


203 


At  a,  S  =  0.905,  the  speed  is  stable.  At  6,  however,  S  =  0.35, 
the  conditions  are  unstable,  and  the  motor  thus  can  not  run  at  6, 
but  either — if  the  speed  should  drop  or  the  load  rise  ever  so  little 
—the  motor  begins  to  slow  down,  thereby,  on  curve,  D,  its  torque 
falls  below  that  of  the  load,  D',  thus  it  slows  down  still  more,  and 
so,  with  increasing  rapidity  the  motor  comes  to  a  standstill.  Or, 
if  the  motor  speed  should  be  a  little  higher,  or  the  load  momen- 
tarily a  little  lower,  the  motor  speed  rises,  until  stability  is  reached 
at  point  a. 


\ 


.7 


FIG.  102. 

With  increasing  load  torque,  D',  the  speed  gradually  drops, 
from  S  =  0.905  at  D'  =  10,  point  a,  down  to  point  c,  at  S  = 
0.75,  D'  =  14.3;  from  there,  however,  the  speed  suddenly  drops 
to  standstill,  that  is,  it  is  not  possible  to  operate  the  motor  at 
speeds  less  than  S  =  0.75,  at  constant  load-torque,  and  the 
branch  of  the  motor  characteristic  from  the  starting  point,  g,  up  to 
the  maximum  torque  point,  c,  is  unstable  on  a  load  requiring  con- 
stant torque. 

At  load  torque,  D'  =  10,  the  motor  can  not  start  the  load,  can 
not  carry  it  below  b,  S  =  0.35 ;  at  speeds  from  b  to  a,  S  =  0.35  to 
0.905,  the  motor  speeds  up;  at  speeds  above  a,  S  =  0.905,  the 
motor  slows  down,  and  drops  into  stable  condition  at  a. 


204  ELECTRIC  CIRCUITS 

With  a  load  torque,  ZX0  =  5,  the  motor  starts  and  runs  up  to 
speed  01,  S  =  0.96. 

D'  =  7.2,  point  g,  thus,  is  the  maximum  load  torque  which  the 
motor  can  start. 

103.  Suppose  now,  while  running  in  stable  condition,  at  point 
a,  with  the  load  torque,  Dr  =  10,  the  load  torque  is  momentarily 
increased.  If  this  increase  leaves  Df  lower  than  the  maximum 
motor  torque,  DO  =  14.3,  the  motor  speed  slows  down,  but  re- 
mains above  c,  and  thus  when  the  increase  of  load  is  taken  off,  the 
motor  again  speeds  up  to  a. 

If,  however,  the  temporary  increase  of  load  torque  exceeds  the 
maximum  motor  torque,  DQ  =  14.3 — for  instance  by  starting  a 
line  of  shafting  or  other  mass  of  considerable  momentum — then 
the  motor  speed  continues  to  drop  as  long  as  the  excess  load  exists, 
and  whether  the  motor  will  recover  when  the  excess  load  is  taken 
off,  or  not,  depends  on  the  loss  of  speed  of  the  motor  during  the 
period  of  overload:  if,  when  the  overload  is  relieved,  the  motor 
has  dropped  to  point  di  in  Fig.  102,  its  speed  thus  is  still  above  b, 
the  motor  recovers;  if,  however,  its  speed  has  dropped  to  dz,  be- 
low the  speed  b,  S  =  0.35,  at  which  the  motor  torque  drops  below 
the  load  torque,  then  the  motor  does  not  recover,  but  stops. 

With  a  lighter  load  torque,  D'0,  which  is  less  than  the  starting 
torque,  g,  obviously  the  motor  will  always  recover  in  speed 

The  amount,  by  which  the  motor  drops  in  speed  at  temporary 
overload,  naturally  depends  on  the  duration  of  the  overload, 
and  on  the  momentum  of  the  motor  and  its  moving  masses: 
the  higher  the  momentum  of  the  motor  and  of  the  masses  driven 
by  it  at  the  moment  of  overload,  the  slower  is  the  drop  of  speed 
of  the  motor,  and  the  higher  thus  the  speed  retained  by  it  at  the 
moment  when  the  overload  is  relieved. 

Thus  a  motor  of  low  starting  torque,  that  is,  high  speed  regula- 
tion, may  be  thrown  out  of  step  by  picking  up  a  load  of  high 
momentum  rapidly,  while  by  adding  a  flywheel  to  the  motor,  it 
would  be  enabled  to  pick  up  this  load.  Or,  it  may  be  troublesome 
to  pick  up  the  first  load  of  high  momentum,  while  the  second  load 
of  this  character  may  give  no  trouble,  as,  due  to  the  momentum 
of  the  load  already  picked  up,  the  speed  would  drop  less. 

Thus  a  motor  carrying  no  load,  may  be  thrown  out  of  step  by  a 
load  which  the  same  motor,  already  partly  loaded  (with  a  load  of 
considerable  momentum),  would  find  no  difficulty  to  pick  up. 

The  ability  of  an  induction  motor,  to  carry  for  a  short  time 


INSTABILITY  OF  CIRCUITS  205 

without  dropping  out  of  step  a  temporary  excessive  overload, 
naturally  also  depends  on  the  excess  of  the  maximum  motor  torque 
(at  c  in  Fig.  102)  over  the  normal  load  torque  of  the  motor.  A 
motor,  in  which  the  maximum  torque  is  very  much  higher — 
several  hundred  per  cent. — than  the  rated  torque,  thus  could 
momentarily  carry  overloads  which  a  motor  could  not  carry, 
in  which  the  maximum  torque  exceeds  the  rated  torque  only  by 
50  per  cent.,  as  was  the  case  with  the  early  motors.  However, 
very  high  maximum  torque  means  low  internal  reactance  and  thus 
high  exciting  current,  that  is,  low  power-factor  at  partial  loads, 
and  of  the  two  types  of  motors: 

(a)  High  overload  torque,  but  poor  power-factor  and  efficiency 
at  partial  loads; 

(6)    Moderate  overload  torque,  but   good  power-factor  and 

efficiency  at  partial  loads; 

the  type  (6)  gives  far  better  average  operating  conditions,  except 
in  those  rare  cases  of  operation  at  constant  full-load,  and  is  there- 
fore preferable,  though  a  greater  care  is  necessary  to  avoid  mo- 
mentary excessive  overloads. 

Gradually  the  type  (a)  had  more  and  more  come  into  use,  as 
the  customers  selected  the  motor,  and  the  power  supply  company 
neglected  to  pay  much  attention  to  power-factor,  and  it  is  only 
in  the  last  few  years,  that  a  realization  of  the  harmful  effects  of 
low  power-factors  on  the  economy  of  operation  of  the  systems  is 
again  directing  attention  to  the  need  of  good  power-factors  at 
partial  loads,  and  the  industry  thus  is  returning  to  type  (6), 
especially  in  view  of  the  increasing  tendency  toward  maximum 
output  rating  of  apparatus. 

In  distributing  transformers,  the  corresponding  situation  had 
been  realized  by  the  central  stations  since  the  early  days,  and 
good  partial  load  efficiencies  and  power-factors  secured. 

104.  The  induction  motor  speed-torque  curve  thus  has  on  a 
constant-torque  load  a  stable  branch,  from  the  maximum  torque 
point,  c,  Fig.  102,  to  synchronism;  and  an  unstable  branch,  from 
standstill  to  the  maximum  torque  point. 

However,  it  would  be  incorrect  to  ascribe  the  stability  or  in- 
stability to  the  induction  motor-speed  curve;  but  it  is  the  char- 
acter of  the  load,  the  requirement  of  constant  torque,  which 
makes  a  part  of  the  speed  curve  unstable,  and  on  other  kinds 
of  load  no  instability  may  exist,  or  a  different  form  of  instability. 

Thus,  considering  a  load  requiring  a  torque  proportional  to 


206 


ELECTRIC  CIRCUITS 


the  speed,  such  as  would  be  given,  approximately,  by  an  electric 
generator  at  constant  field  excitation  and  constant  resistance  as 
load. 

The  load-torque  curves,  then,  would  be  straight  lines  going 
through  the  origin,  as  shown  by  D'i,  D'2,  D'3,  etc.,  for  increasingly 
larger  values  of  load,  in  Fig.  103.  The  motor-torque  curve,  D,  is 
the  same  as  in  Fig.  102.  As  seen,  all  the  lines,  D',  intersect  D  at 
points,  «i,  a2,  a3  .  .  . ,  at  which  the  speed  is  stable,  since 


/D, 


d. 


L 


FIG.  103. 


dS 


dD 

dS' 


Thus,  with  this  character  of  load,  a  torque  required  propor- 
tional to  the  speed,  and  the  motor-torque  curve,  D,  no  instability 
exists,  but  conditions  are  stable  from  standstill  to  synchronism, 
just  as  in  Fig.  101.  That  is,  with  increasing  load,  the  speed  de- 
creases and  increases  again  with  decreasing  load. 

If,  however,  the  motor  curve  is  as  shown  by  D0  in  Fig.  103,  that 
is,  low  starting  torque  and  a  maximum  torque  point  close  to 
synchronism,  as  corresponds  to  an  induction  motor  with  low 
resistance  secondary,  then  for  a  certain  range  of  load,  between 


INSTABILITY  OF  CIRCUITS  207 

D'  and  D'o,  the  load-torque  line,  D'2,  intersects  the  motor  curve, 
Do,  in  three  points  b2,  dz,  h2. 

At  62,  S  =  0.925,  and  at  h2,  S  =  0.375,  conditions  are  stable; 
at  d2,  S  =  0.75,  instability  exists. 

Thus  with  this  load,  D'2,  the  motor  can  run  at  two  different  speeds 
in  stable  conditions:  a  high  speed,  above  c0,  and  a  low  speed,  be- 
low 6;  while  there  is  a  third,  theoretical  speed,  dz,  which  is  unstable. 
In  the  range  below  h2,  the  motor  speeds  up  to  A2;  in  the  range 
between  h2  and  d2)  the  motor  slows  down  to  h2;  in  the  range 
between  d2  and  62,  the  motor  speeds  up  to  62,  and  in  the  range 
above  62,  the  motor  slows  down  to  b2. 

There  is  thus  a  (fairly  narrow)  range  of  loads  between  D'  and 
D'0,  in  which  an  unstable  branch  of  the  induction  motor-torque 
curve  exists,  at  intermediate  speeds;  at  low  speed  as  well  as  at 
high  speed  conditions  are  stable. 

For  loads  less  than  Df,  conditions  are  stable  over  the  entire 
range  of  speed;  for  loads  above  D'o,  the  motor  can  run  only  at  low 
speeds,  h^h^  but  not  at  high  speeds;  but  there  is  no  load  at  which 
the  motor  would  not  start  and  run  up  to  some  speed. 

Obviously,  at  the  lower  speeds,  the  current  consumed  by  the 
motor  is  so  large,  that  the  operation  would  be  very  inefficient. 

It  is  interesting  to  note,  that  with  this  kind  of  load,  the  "maxi- 
mum torqtfe  point,'7  c,  is  no  characteristic  point  of  the  motor- 
torque  curve,  but  two  points,  c0  and  6,  exist,  between  which  the  op- 
eration of  the  motor  is  unstable,  and  the  speed  either  drops  down 
below  6,  or  rises  above  c0. 

105.  With  a  load  requiring  a  torque  proportional  to  the  square 
of  the  speed,  such  as  a  fan,  or  a  ship  propeller,  conditions  are  al- 
most always  stable  over  the  entire  range  of  speed,  from  standstill 
to  synchronism,  and  an  unstable  range  of  speed  may  occur  only  in 
motors  of  very  low  secondary  resistance,  in  which  the  drop  of 
torque  below  the  maximum  torque  point,  c,  of  the  motor  character- 
istic is  very  rapid,  that  is,  the  torque  of  the  motor  decreases  more 
rapidly  than  with  the  square  of  the  speed.  This  may  occur  with 
very  large  motors,  such  as  used  on  ship  propellers,  if  the  secondary 
resistance  is  made  too  low. 

More  frequently  instability  with  such  fan  or  propeller  load  or 
other  load  of  similar  character  may  occur  with  single-phase 
motors,  as  in  these  the  drop  of  the  torque  curve  below  maximum 
torque  is  much  more  rapid,  and  often  a  drop  of  torque  with  in- 
creasing speed  occurs,  especially  with  the  very  simple  and  cheap 


208  ELECTRIC  CIRCUITS 

starting  devices  economically  required  on  very  small  motors,  such 
as  fan  motors. 

Instability  and  dropping  out  of  step  of  induction  motors  also 
may  be  the  result  of  the  voltage  drop  in  the  supply  lines,  and 
furthermore  may  result  from  the  regulation  of  the  generator  vol- 
tage being  too  slow.  Regarding  hereto,  however,  see  "  Theo^  and 
Calculation  of  Electrical  Apparatus,  "in  the  chapter  on  "Stability 
of  Induction  Machines." 

D.  Hunting  of  Synchronous  Machines 

106.  In  induction-motor  circuits,  instability  almost  always 
assumes  the  form  of  a  steady  change,  with  increasing  rapidity, 
from  the  unstable  condition  to  a  stable  condition  or  to  stand- 
still, etc. 

Oscillatory  instability  in  induction-motor  circuits,  as  the  result 
of  the  relation  of  load  to  speed  and  electric  supply,  is  rare.  It 
has  been  observed,  especially  in  single-phase  motors,  in  cases  of 
considerable  oversaturation  of  the  magnetic  circuit. 

Oscillatory  instability,  however,  is  typical  of  the  synchronous 
machine,  and  the  hunting  of  synchronous  machines  has  probably 
been  the  first  serious  problem  of  cumulative  oscillations  in  electric 
circuits,  and  for  a  long  time  has  limited  the  industrial  use  of  syn- 
chronous machines,  in  its  different  forms: 

(a)  Difficulty  and  failure  of  alternating-current  generators  to 
operate  in  parallel. 

(6)  Hunting  of  synchronous  converters. 

(c)  Hunting  of  synchronous  motors. 

While  considerable  theoretical  work  has  been  done,  practically 
all  theoretical  study  of  the  hunting  of  synchronous  machines 
has  been  limited  to  the  calculation  of  the  frequency  of  the  transi- 
ent oscillation  of  the  synchronous  machine,  at  a  change  of  load, 
frequency  or  voltage,  at  synchronizing,  etc.  However,  this 
transient  oscillation  is  harmless,  and  becomes  dangerous  only  if 
the  oscillation  ceases  to  be  transient,  but  becomes  permanent  and 
cumulative,  and  the  most  important  problem  in  the  study  of  hunt- 
ing thus  is  the  determination  of  the  cause,  which  converts  the 
transient  oscillation  into  a  cumulative  one,  that  is,  the  determina- 
tion of  the  source  of  the  energy,  and  the  mechanism  of  its  trans- 
fer to  the  oscillating  system.  To  design  synchronous  machines, 
so  as  to  have  no  or  very  little  tendency  to  hunting,  obviously  re- 


INSTABILITY  OF  CIRCUITS  209 

quires  a  knowledge  of  those  characteristics  of  design  which  are 
instrumental  in  the  energy  transfer  to  the  oscillating  system,  and 
thereby  cause  hunting,  so  as  to  avoid  them  and  produce  the  great- 
est possible  inherent  stability. 

If,  in  an  induction  motor  running  loaded,  at  constant  speed,  the 
load  is  suddenly  decreased,  the  torque  of  the  motor  being  in  ex- 
cess of  the  reduced  load  causes  an  acceleration,  and  the  speed  in- 
creases. As  in  an  induction  motor  the  torque  is  a  function  of  the 
speed,  the  increase  of  speed  decreases  the  torque,  and  thereby  de- 
creases the  increase  of  speed  until  that  speed  is  reached  at  which 
the  motor  torque  has  dropped  to  equality  with  the  load,  and 
thereby  acceleration  and  further  increase  of  speed  ceases,  and  the 
motor  continues  operation  at  the  constant  higher  speed,  that  is, 
the  induction  motor  reacts  on  a  decrease  of  load  by  an  increase  of 
speed,  which  is  gradual  and  steady  without  any  oscillation. 

If,  in  a  synchronous  motor  running  loaded,  the  load  is  suddenly 
decreased,  the  beginning  of  the  phenomenon  is  the  same  as  in 
the  induction  motor,  the  excess  of  motor  torque  causes  an  ac- 
celeration, that  is,  an  increase  of  speed.  However,  in  the 
synchronous  motor  the  torque  is  not  a  function  of  the  speed,  but 
in  stationary  condition  the  speed  must  always  be  the  same, 
synchronism,  and  the  torque  is  a  function  of  the  relative  position 
of  the  rotor  to  the  impressed  frequency.  The  increase  of  speed, 
due  to  the  excess  torque  resulting  from  the  decreased  load,  causes 
the  rotor  to  run  ahead  of  its  previous  relative  position,  and  thereby 
decreases  the  torque  until,  by  the  increased  speed,  the  motor 
has  run  ahead  from  the  relative  position  corresponding  to  the  pre- 
vious load,  to  the  relative  position  corresponding  to  the  decreased 
load.  Then  the  acceleration,  and  with  it  the  increase  of  speed, 
stops.  But  the  speed  is  higher  than  in  the  beginning,  that  is,  is 
above  synchronism,  and  the  rotor  continues  to  run  ahead,  the 
torque  continues  to  decrease,  is  now  below  that  required  by  the 
load,  and  the  latter  thus  exerts  a  retarding  force,  decreases  the 
speed  and  brings  it  back  to  synchronism.  But  when  synchron- 
ous speed  is  reached  again,  the  rotor  is  ahead  of  its  proper  position, 
thus  can  not  carry  its  load,  and  begins  to  slow  down,  until  it  is 
brought  back  into  its  proper  position.  At  this  position,  however, 
the  speed  is  now  below  synchronism,  the  rotor  thus  continues  to 
drop  back,  and  the  motor  torque  increases  beyond  the  load, 
thereby  accelerates  again  to  synchronous  speed,  etc.,  and  in  this 
manner  conditions  of  synchronous  speed,  with  the  rotor  position 

14 


210  ELECTRIC  CIRCUITS 

behind  or  ahead  of  the  position  corresponding  to  the  load,  alter- 
nate with  conditions  of  proper  relative  position  of  the  rotor,  but 
below  or  above  synchronous  speed,  that  is,  an  oscillation  results 
which  usually  dies  down  at  a  rate  depending  on  the  energy  losses 
resulting  from  the  oscillation. 

107.  As  seen,  the  characteristic  of  the  synchronous  machine 
is,  that  readjustment  to  a  change  of  load  requires  a  change  of 
relative  position  of  the  rotor  with  regard  to  the  impressed  fre- 
quency, without  any  change  of  speed,  while  a  change  of  relative 
position  can  be  accomplished  only  by  a  change  of  speed,  and  this 
results  in  an  over-reaching  in  position  and  in  speed,  that  is,  in  an 
oscillation. 

Due  to  the  energy  losses  caused  by  the  oscillation,  the  success- 
ive swings  decrease  in  amplitude,  and  the  oscillation  dies  down. 
If,  however,  the  cause  which  brings  the  rotor  back  from  the  posi- 
tion ahead  or  behind  its  normal  position  corresponding  to  the 
changed  load  (excess  or  deficiency  of  motor  torque  over  the 
torque  required  by  the  load)  is  greater  than  the  torque  which 
opposes  the  deviation  of  the  rotor  from  its  normal  position,  each 
swing  tends  to  exceed  the  preceding  one  in  amplitude,  and  if  the 
energy  losses  are  insufficient,  the  oscillation  thus  increases  in 
amplitude  and  becomes  cumulative,  that  is,  hunting. 

In  Fig.  104  is  shown  diagrammatically  as  p,  the  change  of  the 
relative  position  of  the  rotor,  from  pi  corresponding  to  the  pre- 
vious load  to  pz  the  position  further  forward  corresponding  to  the 
decreased  load. 

v  then  shows  the  oscillation  of  speed  corresponding  to  the 
oscillation  of  position. 

The  dotted  curve,  Wi,  then  shows  the  energy  losses  resulting 
from  the  oscillation  of  speed  (hysteresis  and  eddies  in  the  pole 
faces,  currents  in  damper  windings),  that  is,  the  damping  power, 
assumed  as  proportional  to  the  square  of  the  speed. 

If  there  is  no  lag  of  the  synchronizing  force  behind  the  position 
displacement,  the  synchronizing  force,  that  is,  the  force  which 
tends  to  bring  the  rotor  back  from  a  position  behind  or  ahead  of 
the  position  corresponding  to  the  load,  would  be — or  may  ap- 
proximately be  assumed  as — proportional  to  the  position  dis- 
placement, p,  but  with  reverse  sign,  positive  for  acceleration  when 
p  is  negative  or  behind  the  normal  position,  negative  or  retarding 
when  p  is  ahead.  The  synchronizing  power,  that  is,  the  power 
exerted  by  the  machine  to  return  to  the  normal  position,  then  is 


INSTABILITY  OF  CIRCUITS 


211 


derived  by  multiplying  —  p  with  v,  and  is  shown  dotted  as  wz 
in  Fig.  104.  As  seen,  it  has  a  double-frequency  alternation  with 
zero  as  average. 

The  total  resultant  power  or  the  resulting  damping  effect 
which  restores  stability,  then,  is  the  sum  of  the  synchronizing 
power  Wz  and  the  damping  power  w\,  and  is  shown  by  the  dotted 


\J/ 


w 


FIG.  104. 

curve  w.     As  seen,  under  the  assumption  or  Fig.  104,  in  this  case 
a  rapid  damping  occurs. 

If  the  damping  winding,  which  consumes  a  part  of  all  the  power, 
Wi,  is  inductive — and  to  a  slight  extent  it  always  is — the  current 
in  the  damping  winding  lags  behind  the  e.m.f.  induced  in  it  by 
the  oscillation,  that  is,  lags  behind  the  speed,  v.  The  power,  Wi, 


212  ELECTRIC  CIRCUITS 

or  that  part  of  it  which  is  current  times  voltage,  then  ceases  to  be 
continuously  negative  or  damping,  but  contains  a  positive  period, 
and  its  average  is  greatly  reduced,  as  shown  by  the  drawn  curve, 
Wi,  in  Fig.  104,  that  is,  inductivity  of  the  damper  winding  is  very 
harmful,  and  it  is  essential  to  design  the  damper  winding  as  non- 
inductive  as  possible  to  give  efficient  damping. 

With  the  change  of  position,  p,  the  current,  and  thus  the  ar- 
mature reaction,  and  with  it  the  magnetic  flux  of  the  machine, 
changes.  A  flux  change  can  not  be  brought  about  instantly, 
as  it  represents  energy  stored,  and  as  a  result  the  magnetic  flux 
of  the  machine  does  not  exactly  correspond  with  the  position,  pt 
but  lags  behind  it,  and  with  it  the  synchronizing  force,  F,  as 
shown  in  Fig.  104,  lags  more  or  less,  depending  on  the  design  of  the 
machine. 

The  synchronizing  power  of  the  machine,^,  in  the  case  of  a  lag- 
ging synchronizing  force,  F,  is  shown  by  the  drawn  curve,  w2.  As 
seen,  the  positive  ranges  of  the  oscillation  are  greater  than  the 
negative  ones,  that  is,  the  average  of  the  oscillating  synchronizing 
power  is  positive  or  supplying  energy  to  the  oscillating  system, 
which  energy  tends  to  increase  the  amplitude  of  the  oscillation — in 
other  words,  tends  to  produce  cumulative  hunting. 

The  total  resulting  power,  w  —  w\  +  wz,  under  these  condi- 
tions is  shown  by  the  drawn  curve,  w,  in  Fig.  104.  As  seen,  its 
average  is  still  negative  or  energy-consuming,  that  is,  the  oscilla- 
tion still  dies  out,  and  stability  is  finally  reached,  but  the  average 
value  of  w  in  this  case  is  so  much  less  than  in  the  case  above  dis- 
cussed, that  the  dying  out  of  the  oscillation  is  much  slower. 

If  now,  the  damping  power,  w\,  were  still  smaller,  or  the  aver- 
age synchronizing  power,  wz,  greater,  the  average  w  would 
become  positive  or  supplying  energy  to  the  oscillating  system. 
In  other  words,  the  oscillation  would  increase  and  hunting 
result. 

That  is: 

If  the  average  synchronizing  power  resulting  from  the  lag  of 
the  synchronizing  force  behind  the  position  exceeds  the  average 
damping  power,  hunting  results.  The  condition  of  stability  of 
the  synchronous  machine  is,  that  the  average  damping  power  ex- 
ceeds the  average  synchronizing  power,  and  the  more  this  is  the 
case,  the  more  stable  is  the  machine,  that  is,  the  more  rapidly 
the  transient  oscillation  of  readjustment  to  changed  circuit  con- 
ditions dies  out. 


INSTABILITY  OF  CIRCUITS  213 

Or,  if 

a  =  attenuation  constant  of  the  oscillating  system, 
a<0  gives  cumulative  oscillation  or  hunting. 
a>0  gives  stability. 

108.  Counting  the  time,  t,  from  the  moment  of  maximum  back- 
ward position  of  the  rotor,  that  is,  the  moment  at  which  the  load 
on  the  machine  is  decreased,  and  assuming  sinusoidal  variation, 
and  denoting 

0  =  2  Trft  =  at  (1) 

where 

/  =  frequency  of  the  oscillation  (2) 

the  relative  position  of  the  rotor  then  may  be  represented  by 

p  =  —  poe00  cos  <£, 
where 

p0  =  p2  —  pl  '=  position  difference  of  rotor  resulting  from 
change  of  load,  (3) 

a  =  attenuation  constant  of  oscillation.  (4) 

The  velocity  difference  from  that  of  uniform  rotation  then  is 

os0).  (5) 


(6) 


sin  a  =  -jj     cos  a  =  -r  (7) 

-i     L  JC\. 

it  is 

v  =  copoAe"00  sin  (<f>  -f-  a).  (8) 

Let 

7  =  lag  of  damping  currents  behind  e.m.f.   induced  in 
damper  windings  (9) 

the  damping  power  is 

Wi  =  —  cwy 

where 

c  =  —^  =  damping  power  per  unit  velocity  and  vy  is  v, 
lagged  by  angle  7.  (11) 


dp 
V=dt 

.«fea 

d<f> 

=    (Op0€~a 

*  (sin  0  4 

Let 

a  =  tan 

«;    l  + 

a2  =  A2 

hence, 

a 

1 

214  ELECTRIC  CIRCUITS 

Let 

0  =  lag  of  synchronizing  force  behind  position  displace- 
ment p  (12) 
and 

0  =  co/o  (13) 

where 

to  =  time  lag  of  synchronizing  force.  (14) 

The  synchronizing  force  then  is 

F  =  bp0e-a*  cos  (0  -  0)  (15) 

where 

ET 

b  =  —  =  ratio  of  synchronizing  force  to  po- 
sition displacement,  or  specific  synchronizing  force.  (16) 
'  The  synchronizing  power  then  is 

wz  =  Fv  =  bwp0Ae-2a+  sin  (0  +  a)  cos  (0  -  0).  (17) 

The  oscillating  mechanical  power  is 

d  mv2  dv 

w  =  -T. =  mwv  -T- 

d*    e  d0 

=  mw@pQ2A26~2  a*  sin  (0  +  «) 

{cos  (0  +  a)  -  a  sin  (0  +  a)}       (18) 
where 

m  =  moving  mass  reduced  to  the  radius,  on 

which  p  is  measured.  (19) 

It  is,  however, 

u>i  +  w2  —  w  =  0  (20) 

hence,   substituting   (10),    (17),    (18)   into    (20)    and   canceling, 
b  cos  (0  —  0)  —  ccoA  sin  (0  +  a  —  7)  — 

mco2Acos  (0  +  a)  +  mw2Aa  sin  (0  +  a)  =  0.      (21) 
This  gives,  as  the  coefficients  of  cos  0  and  sin  0  the  equations 

6  cos  0  —  cuA  sin  (a  —  7)  —  ma2 A  cos  a  -h  mu2Aasm  a  =  0       ,     , 
b  sin  /3  —  ccoA  cos  (a  —  7)  -f  mu2A  sin  a  +  raa>2A  cos  a  =  0 

Substituting  (6)  and  (7)  and  approximating  from  (13),  for 
0  as  a  small  quantity, 

cos/3  =  1;     sin/3  =  o>£0  (23) 

gives 

6  —  ceo  (  a  cos  7  —  sin  7)  —  mu2  (1  —  a2)  =  0          ,_,. 
fao  —  c  (cos  7  +  a  sin  7)  -f-  2wa>a  =  0 


INSTABILITY  OF  CIRCUITS  215 

This  gives  the  values,  neglecting  smaller  quantities 

a  =  -         ccos7-fr<o  (2g) 

\/4  mb  -  c2  cos2  T  +  W 

<o  =  —  {  V4  w&  -  c2  cos  2  7  +  6V  +  csiny}          (26) 

Zl7l 

f  =  ft  (27> 

These  equations  (25)  and  (26)  apply  only  for  small  values  of 
a,  but  become  inaccurate  for  larger  values  of  a,  that  is,  very  rapid 
damping.  However,  the  latter  case  is  of  lesser  importance. 

a  =  0 
gives 

bt0  =  c  cos  7, 
hence, 


cos  7 
or, 

c  cos  7 


are  the  conditions  of  stability  of  the  synchronous  machine. 
If 

to  =  0 

7  =  0 
it  is 

c 


(28) 


a  = 


V4  mb  -  c2 

\/4  ra&  —  c 

<*)  = 

2  m 

and,  if  also, 

c  =  0: 
it  is 


CHAPTER  XII 
REACTANCE  OF  INDUCTION  APPARATUS 

109.  An  electric  current  passing  through  a  conductor  is  ac- 
companied by  a  magnetic  field  surrounding  this  conductor,  and 
this  magnetic  field  is  as  integral  a  part  of  the  phenomenon,  as  is 
the  energy  dissipation  by  the  resistance  of  the  conductor.  It  is 
represented  by  the  inductance,  L,  of  the  conductor,  or  the  number 
of  magnetic  interlinkages  with  unit  current  in  the  conductor. 
Every  circuit  thus  has  a  resistance,  and  an  inductance,  however 
small  the  latter  may  be  in  the  so-called  "non-inductive"  circuit. 
With  continuous  current  in  stationary  conditions,  the  inductance, 
L,  has  no  effect  on  the  energy  flow;  with  alternating  current  of 
frequency,  /,  the  inductance,  L,  consumes  a  voltage  2  irfLi,  and  is, 
therefore,  represented  by  the  reactance,  x  =  2-jrfL,  which  is 
measured  in  ohms,  and  differs  from  the  ohmic  resistance,  r,  merely 
by  being  wattless  or  reactive,  that  is,  representing  not  dissipation 
of  energy,  but  surging  of  energy. 

Every  alternating-current  circuit  thus  has  a  resistance  and  a 
reactance,  the  latter  representing  the  effect  of  the  magnetic  field 
of  the  current  in  the  conductor. 

When  dealing  with  alternating-current  apparatus,  especially 
those  having  several  circuits,  it  must  be  realized,  however,  that 
the  magnetic  field  of  the  circuit  may  have  no  independent  exist- 
ence, but  may  merge  into  and  combine  with  other  magnetic  fields, 
so  that  it  may  become  difficult  what  part  of  the  magnetic  field  is 
to  be  assigned  to  each  electric  circuit,  and  circuits  may  exist 
which  apparently  have  no  reactance.  In  short,  in  such  cases, 
the  magnetic  fields  of  the  reactance  of  the  electric  circuit  may  be 
merely  a  more  or  less  fictitious  component  of  the  resultant  mag- 
netic field. 

The  industrial  importance  hereof  is  that  many  phenomena,  such 
as  the  loss  of  power  by  magnetic  hysteresis,  the  m.m.f.  required 
for  field  excitation,  etc.,  are  related  to  the  resultant  magnetic 
field,  thus  not  equal  to  the  sum  of  the  corresponding  effects  of  the 
components. 

216 


REACTANCE  OF  INDUCTION  APPARATUS         217 

As  the  transformer  is  the  simplest  alternating-current  appara- 
tus, the  relations  are  best  shown  thereon. 

Leakage  Flux  of  Alternating-current  Transformer 

110.  The  alternating-current  transformer  consists  of  a  mag- 
netic circuit,  interlinked  with  two  electric  circuits,  the  primary 
circuit,  which  receives  power  from  its  impressed  voltage,  and 
the  secondary  circuit,  which  supplies  power  to  its  external  circuit. 

For  convenience,  we  may  assune  the  secondary  circuit  as  re- 
duced to  the  primary  circuit  by  the  ratio  of  turns,  that  is,  assume 
ratio  of  turns  1  -j-  1. 
Let 

YQ  =  g  —  jb  —  primary  exciting  admittance; 

ZQ  =  r0  +  jxQ  =  primary  self-inductive  impedance; 

Zi  =  ri  +  jxi  =  secondary  self-inductive  impedance  (reduced 
to  the  primary). 

The  transformer  thus  comprises  three  magnetic  fluxes:  the 
mutual  magnetic  flux,  $,  which,  being  interlinked  with  primary 
and  secondary,  transforms  the  power  from  primary  to  secondary, 
and  is  due  to  the  resultant  m.m.f  of  primary  and  secondary  cir- 
cuit; the  primary  leakage  flux,  $'0,  due  to  the  m.m.f.  of  the  primary 
circuit,  FQ,  and  interlinked  with  the  primary  circuit  only,  which  is 
represented  by  the  self-inductive  or  leakage  reactance,  x0;  and  the 
secondary  leakage  flux,  3>'i,  due  to  the  m.m.f.  of  the  secondary 
circuit,  FI,  and  interlinked  with  the  secondary  circuit  only 
which  is  represented  by  the  secondary  reactance,  Xi. 

As  seen  in  Fig.  105o,  the  mutual  flux,  $ — usually — has  a  closed 
iron  circuit  of  low  reluctance,  p,  thus  low  m.m.f.,  F,  and  high  intens- 
ity; the  self-inductive  flux  or  leakage  reactance  flux,  3>'o  and  <l>'i, 
close  through  the  air  circuit  between  the  primary  and  secondary 
electric  circuits,  thus  meet  with  a  high  reluctance,  po,  respectively 
Pi,  usually  many  hundred  times  higher  than  p.  Their  m.m.fs.,  FQ 
and  Fij  however,  are  usually  many  times  greater  than  F\  the  lat- 
ter is  the  m.m.f.  of  the  exciting  current,  the  former  that  of  full 
primary  or  secondary  current. 

For  instance,  if  the  exciting  current  is  5  per  cent,  of  full-load 
current,  the  reactance  of  the  transformer  4  per  cent.,  or  2  per  cent, 
primary  and  2  per  cent,  secondary,  then  the  m.m.f.  of  the  leakage 
flux  is  20  times  that  of  the  mutual  flux,  and  the  mutual  flux  50 
times  the  leakage  flux,  hence  the  reluctance  of  leakage  flux  50 
X  20  =  1000  times  that  of  the  mutual  or  main  flux:  pi  =  1000  p. 


218 


ELECTRIC  CIRCUITS 


•e    *'M 


I*;  5| 


I  *;  5 1 


/ 


S    P 


In  HI 


S    P 


I  *v  .*«.! 


S    P 


I 


S    P 


FIG.  105. 


REACTANCE  OF  INDUCTION  APPARATUS 


219 


111.  Usually,  as  stated,  the  leakage  fluxes  are  not  considered 
as  such,  but  represented  by  their  reactances,  in  the  transformer 
diagram.  Thus,  at  non-inductive  load,  it  is,  Fig.  106, 

0$  =  mutual,  or  main  magnetic  flux,  chosen  as  negative  ver- 
tical. 

OF  =  m.m.f.  required  to  produce  flux,  0<J>,  and  leading  it  by  the 

angle  of  hysteretic  advance  of  phase,  F0$. 
OE\  =  e.m.f.  induced  in  the  secondary  circuit  by  the  mutual  flux, 
and  90°  behind  it. 


TRANSFORMER  DIAGRAM 

NON-INDUCTIVE  LOAD 
SHOWING  MAGNETIC  FLUXES 


Fo 


FIG.  106. 

IiXi  =  secondary  reactance  voltage,  90°  behind  the  secondary 
current,  and  combining  with  OE'i  to 

OEi  =  true  secondary  induced  voltage.  From  this  subtracts 
the  secondary  resistance  voltage,  Jiri,  leaving  the  sec- 
ondary terminal  voltage,  and,  in  phase  with  it  at  non- 
inductive  load,  the  secondary  current  and  secondary 
m.m.f.,  OF i. 

From  component,  OF\,  and  resultant,  OF,  follows  the  other  com- 
ponent, 


220  ELECTRIC  CIRCUITS 

OF0    =  primary  m.m.f.  and  in  phase  with  it  the  primary 

current. 
OEfQ  =  primary    voltage    consumed    by    mutual    flux, 

equal  and  opposite  to  OE'\. 
loXo    =  primary  reactance  voltage,   90°   ahead   of   the 

primary  current  OFQ. 

From  IQXO  as  component  and  E'o  as  resultant  follows  the  other 
component,  OEQ,  and  adding  thereto  the  primary  resistance  vol- 
tage, IQTQ,  gives  primary  supply  voltage. 

In  this  diagram,  Fig.  106,  the  primary  leakage  flux  is  represented 
by  0<l>'o,  in  phase  with  the  primary  current,  OF0,  and  the  secondary 
leakage  flux  is  represented  by  O&'i,  in  phase  with  the  secondary 
current,  OF\. 

As  shown  in  Fig.  105o,  the  primary  leakage  flux,  $'0,  passes 
through  the  iron  core  inside  of  the  primary  coil,  together  with  the 
resultant  flux,  3>,  and  the  secondary  leakage  flux,  3>'i,  passes  through 
the  secondary  core,  together  with  the  mutual  flux,  3>.  However, 
at  the  moment  shown  in  Fig.  105o,  3>'i  and  <£  in  the  secondary 
core  are  opposite  in  direction.  This  obviously  is  not  possible, 
and  the  flux  in  the  secondary  core  in  this  moment  is  $  —  $'i, 
that  is,  the  magnetic  disposition  shown  in  Fig.  1050  is  merely 
nominal,  but  the  actual  magnetic  distribution  is  as  shown  in 
Fig.  105a;  the  flux  in  the  primary  core,  $o  =  $  +  $'o,  the  flux  in 
the  secondary  core,  $1  =  $  —  3>'i. 

As  seen,  at  the  moment  shown  in  Fig.  105o  and  105a,  all  the 
leakage  flux  comes  from  and  interlinks  with  the  primary  winding, 
none  with  the  secondary  winding,  and  it  thus  would  appear,  that 
all  the  self-inductive  reactance  is  in  the  primary  circuit,  none  in 
the  secondary  circuit,  or,  in  other  words,  that  the  secondary 
circuit  of  the  transformer  has  no  reactance. 

However,  at  a  later  moment  of  the  cycle,  shown  in  Fig.  105c, 
all  the  leakage  flux  comes  from  and  interlinks  with  the  secondary, 
and  this  figure  thus  would  give  the  impression,  that  all  the  leakage 
reactance  of  the  transformer  is  in  the  secondary,  none  in  the 
primary  winding. 

In  other  words,  the  leakage  fluxes  of  the  transformer  and  the 
mutual  or  main  flux  are  not  independent  fluxes,  but  partly  tra- 
verse the  same  magnetic  circuit,  so  that  each  of  them  during  a  part 
of  the  cycle  is  a  part  of  any  other  of  the  fluxes.  Thus,  the  react- 
ance voltage  and  the  mutual  inductive  voltage  of  the  transformer 


REACTANCE  OF  INDUCTION  APPARATUS 


221 


are  not  separate  e.m.fs.,  but  merely  mathematical  fictions,  com- 
ponents of  the  resultant  induced  voltage,  OE\  and  OE0,  induced 
by  the  resultant  fluxes,  0$o  in  the  primary,  and  O^i  in  the  sec- 
ondary core. 

112.  In  Fig.  107  are  plotted,  in  rectangular  coordinates,  the 
magnetic  fluxes: 

The  mutual  or  main  magnetic  flux,  $>; 

The  primary  leakage  flux,  3>'0; 

The  resultant  primary  flux,  <J>0  =  3>  +  $'0; 

The  secondary  leakage  flux,  <J>'i; 

The  resultant  secondary  flux,  $1  =  <J>  —  <£'i; 


MAGNETIC  FLUXES  OF 

TRANSFORMER 
0  =  6.2 

0i=1.5  01  =  1.05°;  $1-6 
00  =  1.9  0o  -  60°;  $0  =  7.6. 

FIG.  107. 


and  the  magnetic  distribution  in  the  transformer,  during  the 
moments  marked  as  a,  b,  c,  d,  e,  f,  g,  in  Fig.  107,  is  shown  in 
Fig.  105. 

In  Fig.  105a,  the  primary  flux  is  larger  than  the  secondary, 
and  all  leakage  fluxes  (XQ  and  Xi)  come  from  the  primary  flux, 
that  is,  there  is  no  secondary  leakage  flux. 

In  Fig.  1056,  primary  and  secondary  flux  equal,  and  primary 
and  secondary  leakage  flux  equal  and  opposite,  though  small. 

In  Fig.  105c,  the  secondary  flux  is  larger,  all  leakage  flux  (x0 
and  Xi)  comes  from  the  secondary  flux,  that  is,  there  is  no 
primary  leakage  flux. 


222  ELECTRIC  CIRCUITS 

In  Fig.  105d,  there  is  no  primary  flux,  and  all  the  secondary 
flux  is  leakage  flux. 

In  Fig.  105e,  there  is  no  mutual  flux,  all  primary  flux  is 
primary  leakage  flux,  and  all  secondary  flux  is  secondary  leakage 
flux. 

In  Fig.  105f,  there  is  no  secondary  flux,  and  all  primary  flux 
is  leakage  flux. 

In  Fig.  1050,  the  primary  flux  is  larger  than  the  secondary, 
and  all  leakage  flux  comes  from  the  primary,  the  same  as  in  105a. 

Figs.  105a  to  105/,  thus  show  the  complete  cycle,  corresponding 
to  diagrams,  Figs.  106  and  107. 

These  figures  are  drawn  with  the  proportions, 

P  -*-  PO    *  PI    ~  1  +  12.5      -T-  12.5 
F  -s-  Fo  -s-  Fi  =  1  -5-    3.8      ^3 

$  +  $'„  -T-  $'i  =  1  •*•    0.317  -T-    0.25. 

thus  are  greatly  exaggerated,  to  show  the  effect  more  plainly. 
Actually,  the  relations  are  usually  of  the  magnitude, 

P  -5-  PO  •*•  Pi  =  1  -s-  1000  -f-  1000 
F  -f-  Fo  •*-  Fi  =  1  •*-  20.6  -T-  20 
$  -T-  $'o  -*•  *'i  =  1  -*•  0.02  -v-  0.02 

113.  In  symbolic  representation,  denoting, 

<i>  =  mutual  magnetic  flux. 
E  =  mutual  induced  voltage. 
<l>o=  resultant  primary  flux. 
f>;0  =  primary  leakage  flux. 
$o  =  primary  terminal  voltage. 
/o  =  primary  current. 

ZQ  =  r0  +  jxQ  =  primary  self-inductive  imped- 
ance. 

f  i  =  resultant  secondary  flux. 
<|>'i  =  secondary  leakage  flux. 
.pi  =  secondary  terminal  voltage. 
/i  =  secondary  current. 

Zi  =  TI  +  jrci  =  secondary  self-inductive   im- 
pedance. 
and 

c  =  2irfn 
where  n  =  number  of  turns. 


REACTANCE  OF  INDUCTANCE  APPARATUS     223 
It  then  is 


cf  =  f  =  E0  -  Zo 
cf  o  =  #0  -  r0/o  = 

Cf  1    =   #1  +  ri/i    =   #   -  jZl/l 

f'o  =  f  o  —  $ 

f  'l  =  f  +  f  1, 

thus,  the  total  leakage  flux 

f  =  f'0  -f  <£>'  !  =  f  o  -  f  i. 

114.  One  of  the  important  conclusions  from  the  study  of  the 
actual  flux  distribution  of  the  transformer  is  that  the  distinction 
between  primary  and  secondary  leakage  flux,  <£'0  and  <£'i,  is  really 
an  arbitrary  one.  There  is  no  distinct  primary  and  secondary 
leakage  flux,  but  merely  one  leakage  flux,  $',  which  is  the  flux 
passing  between  primary  and  secondary  circuit,  and  which  during 
a  part  of  the  cycle  interlinks  with  the  primary,  during  another 
part  of  the  cycle  interlinks  with  the  secondary  circuit  Thus  the 
corresponding  electrical  quantities,  the  reactances,  XQ  and  Xi,  are 
not  independent  quantities,  that  is,  it  can  not  be  stated  that  there 
is  a  definite  primary  reactance,  XQ,  and  a  definite  secondary  react- 
ance, Xi,  but  merely  that  the  transformer  has  a  definite  reactance, 
x}  which  is  more  or  less  arbitrarily  divided  into  two  parts  ;  x  =  XQ 
+  £1,  and  the  one  assigned  to  the  primary,  the  other  to  the  second- 
ary circuit. 

As  the  result  hereof,  "  mutual  magnetic  flux"  <£,  and  the  mutual 
induced  voltage,  E,  are  not  actual  quantities,  but  rather  mathe- 
matical fictions,  and  not  definite  but  dependent  upon  the  distri- 
bution of  the  total  reactance  between  the  primary  and  the  sec- 
ondary circuit. 

This  explains  why  all  methods  of  determining  the  transformer 
reactance  give  the  total  reactance  XQ  +  x\. 

However,  the  subdivision  of  the  total  transformer  reactance 
into  a  primary  and  a  secondary  reactance  is  not  entirely  arbitrary. 
Assuming  we  assign  all  the  reactance  to  the  primary,  and  consider 
the  secondary  as  having  no  reactance.  Then  the  mutual  mag- 
netic flux  and  mutual  induced  voltage  would  be 

cf  =  E  =  #o  -  [r0  +  j  (XQ  +  xi)]  h 

and  the  hysteresis  loss  in  the  transformer  would  correspond  hereto, 
by  the  usual  assumption  in  transformer  calculations. 


224  ELECTRIC  CIRCUITS 

Assigning,  however,  all  the  reactance  to  the  secondary  circuit, 
and  assuming  the  primary  as  non-inductive,  the  mutual  flux  and 
mutual  induced  voltage  would  be  c$  =  E  =  EQ  —  r0/0,  hence 
larger,  and  the  hysteresis  loss  calculated  therefrom  larger  than 
under  the  previous  assumption.  The  first  assumption  would 
give  too  low,  and  the  last  too  high  a  calculated  hysteresis  loss,  in 
most  cases. 

By  the  usual  transformer  theory,  the  hysteresis  loss  under  load 
is  calculated  as  that  corresponding  to  the  mutual  induced  voltage, 
E.  The  proper  subdivision  of  the  total  transformer  reactance,  x, 
into  primary  reactance,  z0,  and  secondary  reactance,  xi,  would  then 
be  that,  which  gives  for  a  uniform  magnetic  flux,  $,  corresponding 
to  the  mutual  induced  voltage,  E,  the  same  hysteresis  loss,  as 
exists  with  the  actual  magnetic  distribution  of  $0  =  $  +  $'o  in 
the  primary,  and  $1  =  $  —  <£'i  in  the  secondary  core.  Thus,  if 
VQ  is  the  volume  of  iron  carrying  the  primary  flux,  <£0,  at  flux  den- 
sity, BQ,  Vi  the  volume  of  iron  carrying  the  secondary  flux,  $1,  at 
flux  density,  BI,  the  flux  density  of  the  theoretical  mutual  mag- 
netic flux  would  be  given  by 

so1-6  +  ViBj-* 


B 


1.6 


from  B  then  follows  3>,  E,  and  thus  XQ  and  a?i. 

This  does  not  include  consideration  of  eddy-current  losses. 
For  these,  an  approximate  allowance  may  be  made  by  using  1.7 
as  exponent,  instead  of  1.6. 

Where  the  magnetic  stray  field  under  load  causes  additional 
losses  by  eddy  currents,  these  are  not  included  in  the  loss  assigned 
to  the  mutual  magnetic  flux,  but  appear  as  an  energy  component 
of  the  leakage  reactances,  that  is,  as  an  increase  of  the  ohmic  re- 
sistances of  the  electric  circuits,  by  an  effective  resistance. 

115.  Usually,  the  subdivision  of  x  into  XQ  and  x\,  by  this  as- 
sumption of  assigning  the  entire  core  loss  to  the  mutual  flux, 
is  sufficiently  close  to  equality,  to  permit  this  assumption.  That 
is,  the  total  transformer  reactance  is  equally  divided  between 
primary  and  secondary  circuit. 

This,  however,  is  not  always  justified,  and  in  some  cases,  the 
one  circuit  may  have  a  higher  reactance  than  the  other.  Such, 
for  instance,  is  the  case  in  some  very  high  voltage  transformers, 
and  usually  is  the  case  in  induction  motors  and  similar  apparatus. 

It  is  more  commonly  the  case,  where  true  self-inductive  fluxes 


REACTANCE  OF  INDUCTION  APPARATUS         225 

exist,  that  is,  magnetic  fluxes  produced  by  the  current  in  one 
circuit,  and  interlinked  with  this  circuit,  closing  upon  themselves 
in  a  path  which  is  entirely  distinct  from  that  of  the  mutual  mag- 
netic flux,  that  is,  has  no  part  in  common  with  it.  Such,  for  in- 
stance, frequently  is  the  self-inductive  flux  of  the  end  connections 
of  coils  in  motors,  transformers,  etc.  To  illustrate :  in  the  high- 
voltage  shell-type  transformer,  shown  diagrammatically  in 
Fig.  108,  with  primary  coil  1,  closely  adjacent  to  the  core,  and 
high-voltage  secondary  coil  2  at  considerable  distance: 

The  primary  leakage  flux  consists  of  the  flux  in  spaces,  a, 
between  the  yokes  of  the  transformer,  closing  through  the  iron 
core,  C,  and  the  flux  through  the  spaces,  b,  outside  of  the  trans- 
former, which  enters  the  faces,  F,  of  the  yokes  and  closes  through 
the  central  core,  C. 

The  secondary  leakage  flux  contains  the  same  two  components : 
the  flux  through  the  spaces,  a,  between  the  yokes  closing,  however, 
through  the  outside  shells,  $,  and  the  flux  through  the  spaces,  6, 
outside  of  the  transformer,  and  entering  the  faces,  F,  but  in  this 
case  closing  through  the  shells,  S.  In  addition  to  these  two  com- 
ponents, the  secondary  leakage  flux  contains  a  third  component, 
passing  through  the  spaces,  6,  between  the  coils,  but  closing, 
through  outside  space,  c,  in  a  complete  air  circuit.  This  flux 
has  no  corresponding  component  in  the  primary,  and  the  total 
secondary  leakage  reactance  in  this  case  thus  is  larger  than  the 
total  primary  reactance. 

Similar  conditions  apply  to  magnetic  structures  as  in  the  in- 
duction motor,  alternator,  etc. 

In  such  a  case  as  represented  by  Fig.  108,  the  total  reactance 
of  the  transformer,  with  (2)  as  primary  and  (1)  as  secondary, 
would  be  greater  than  with  (1)  as  primary  and  (2)  as  secondary. 

In  this  case,  when  subdividing  the  total  reactance  into  primary 
reactance  and  secondary  reactance,  it  would  appear  legitimate 
to  divide  it  in  proportion  of  the  total  reactances  with  (1)  and  (2) 
as  primary,  respectively.  That  is, 

if  x  =  total  reactance,  with  coil   (1)   as  primary, 

and  (2)  as  secondary,  and 

x'  =  total  reactance,  with  coil   (2)   as  primary, 
and  (1)  as  secondary,  then  it  is: 

With  coil  (1)  as  primary  and  (2)  as  secondary, 

15 


226  ELECTRIC  CIRCUITS 

Primary  reactance, 


Xo 


Secondary  reactance, 


xx' 


x  +  x'~       x  +  x' 
With  coil  (2)  as  primary  and  (1)  as  secondary, 


Primary  reactance, 


X0   = 


FIG.  108. 


, 

-.X   = 


Secondary  reactance, 


x  +  x 


x  +  x'         x  +  x' 


116.  By  test,  the  two  total  reactances,  x  and  x',  can  be  derived 
by  considering,  that  in  Fig.  107  at  the  moments,  /  and  d,  the  total 
flux  is  leakage  flux,  as  more  fully  shown  in  Fig.  105/  and  105d, 
and  the  flux  measured  from  /,  gives  the  reactance,  x,  measured 
from  d,  gives  the  reactance,  d. 


REACTANCE  OF  INDUCTION  APPARATUS 


227 


Assuming  we  connect  primary  coil  and  secondary  coil  in  series 
with  each  other,  but  in  opposition,  into  an  alternating-current 
circuit,  as  shown  in  Fig.  109,  and  vary  the  number  of  primary 
and  secondary  turns,  until  the  voltage,  e\,  across  the  secondary 
coil,  s,  becomes  equal  to  r-^i.  Then  no  flux  passes  through  the 
secondary  coil,  that  is,  the  condition,  Fig.  107 /,  exists,  and  the 
voltage,  60,  across  the  primary  coil,  p,  gives  the  total  reactance, 
x,  for  p  as  primary, 

eo2  =  i*  (r02  +  z2). 

Varying  now  the  number  of  turns  so  that  the  voltage  across 
the  primary  coil  equals  its  resistance  drop,  eQ  =  r0i,  then  the 


FIG.  109. 


voltage  across  the  secondary  coil,  s,  gives  the  total  reactance, 
for  s  as  primary, 


It  would  rarely  be  possible  to  vary  the  turns  of  the  two  coils, 
p  and  s.  However,  if  we  short-circuit  s  and  pass  an  alternating 
current  through  p,  then  at  the  very  low  resultant  magnetic  flux 
and  thus  resultant  m.m.f.,  primary  and  secondary  current  are 
practically  in  opposition  and  of  the  same  m.m.f.,  and  the  mag- 
netic flux  in  the  secondary  coil  is  that  giving  the  resistance  drop 
riiiy  that  is,  e'i  =  7*1  ii  is  the  true  primary  voltage  in  the  secondary, 
and  the  voltage  across  the  primary  terminals  thus  is  that  giving 
primary  resistance  drop,  r0io,  total  self-inductive  reactance, 
and  the  secondary  induced  voltage,  rii\.  Thus, 


228  ELECTRIC  CIRCUITS 

or,  since  ii  practically  equals  iQ, 

e,2  =  i^  [(r0  +  n)2  +  x*}, 

and  inversely,  impressing  a  voltage  upon  coil,  s,  and  short-cir- 
cuiting the  coil  p,  gives  the  leakage  reactance,  x',  for  s  as  primary, 


Thus,  the  so-called  "impedance  test"  of  the  transformer  gives 
the  total  leakage  reactance  XQ  +  x\t  for  that  coil  as  primary, 
which  is  used  as  such  in  the  impedance  test. 

Where  an  appreciable  difference  of  the  total  leakage  flux  is 
expected  when  using  the  one  coil  as  primary,  as  when  using  the 
other  coil,  the  impedance  tests  should  be  made  with  that  coil  as 
primary,  which  is  intended  as  such.  Since,  however,  the  two 
leakage  fluxes  are  usually  approximately  equal,  it  is  immaterial 
which  coil  is  used  as  primary  in  the  impedance  test,  and  gener- 
ally that  coil  is  used,  which  gives  a  more  convenient  voltage  and 
current  for  testing. 

Magnetic  Circuits  of  Induction  Motor 

117.  In  general,  when  dealing  with  a  closed  secondary  winding, 
as  an  induction-motor  squirrel-cage,  we  consider  as  the  mutual 
inductive  voltage,  E,  the  voltage  induced  by  the  mutual  magnetic 
flux,  $,  that  is,  the  magnetic  flux  due  to  the  resultant  of  the  pri- 
mary and  the  secondary  m.m.f.  This  voltage,  E,  then  is  con- 
sumed in  the  closed  secondary  winding  by  the  resistance,  ri/i, 
and  the  reactance,  jx\J\t  thus  giving,  E  =  (n  +  jxi)  I\. 

The  reactance  voltage,  jx-ij\,  is  consumed  by  a  self-inductive 
flux,  $1',  that  is,  a  magnetic  leakage  flux  produced  by  the  second- 
ary current  and  interlinked  with  the  secondary  circuit,  and  the 
actual  or  resultant  magnetic  flux  interlinked  with  the  secondary 
circuit,  that  is,  the  magnetic  flux,  which  passes  beyond  the  second- 
ary conductor  through  the  armature  core,  thus  is  the  vector  dif- 
ference, <f>i  =  <£>  —  i<i>',  and  the  actual  voltage  induced  in  the  second-  1 
ary  circuit  by  the  resultant  magnetic  flux  interlinked  with  it  thus 
is,  EI  =  1$  —  jxji.  This  voltage  is  consumed  by  the  resistance 
of  the  secondary  circuit,  EI  =  ri/i,  and  the  voltage  consumed  by 
salf-induction,  jx\J\,  is  no  part  of  EI,  but  as  stated,  is  due  to  the 
self-inductive  flux,  <|>i',  which  vectorially  subtracts  from  the 
mutual  magnetic  flux,  <f>,  and  thereby  leaves  the  flux,  <|>i',  which 
induces  1. 


REACTANCE  OF  INDUCTION  APPARATUS         229 

In  other  words: 

In  any  closed  secondary  circuit,  as  a  squirrel-cage  of  an  induc- 
tion motor,  the  true  induced  e.m.f.  in  the  circuit,  that  is,  the  e.m.f. 
induced  by  the  actual  magnetic  flux  interlinked  with  the  circuit, 
is  the  resistance  drop  of  the  circuit,  EI  =  rji. 

This  is  true  whether  there  is  one  or  any  number  of  closed  sec- 
ondary circuits  —  or  squirrel-cages  in  an  induction  motor.  In  each 

Tjl 

the  current,  /i  is  —  l,  where  n  is  the  resistance  of  the  circuit,  and 

$1  the  voltage  induced  by  the  flux  which  passes  through  the  cir- 
cuit. The  fli  of  the  different  squirrel-cages  then  would  differ 
from  each  other  by  the  voltage  induced  by  the  leakage  flux 
which  passes  between  them,  and  which  is  represented  by  the  self- 
inductive  reactance  of  the  next  squirrel-cage: 


K 

where  /'i  =  -7-^  is  the  current  in  the  inner  squirrel-cage  of  voltage, 
T  i 

#'i,  and  resistance,  r'i,  and  a/i/'i,  is  the  reactance  of  the  flux 
between  the  two  squirrel-cages. 

The  mutual  magnetic  flux  and  the  mutual  induced  e.m.f.  of  the 
common  induction  motor  theory  thus  are  mathematical  fictions 
and  not  physical  realities. 

The  advantage  of  the  introduction  of  the  mutual  magnetic 
flux,  $,  and  the  mutual  induced  voltage,  E,  in  the  induction-motor 
theory,  is  the  ease  and  convenience  of  passing  therefrom  to  the 
secondary  as  well  as  the  primary  circuit.  Where,  however,  a 
number  of  secondary  circuits  exist,  as  in  a  multiple  squirrel-cage, 
it  is  preferable  to  start  from  the  innermost  magnetic  flux,  that  is, 
the  magnetic  flux  passing  through  the  innermost  squirrel-cage, 
and  the  voltage  induced  by  it  in  the  latter,  which  is  the  resistance 
drop  of  this  squirrel-cage. 

In  the  same  manner,  in  a  primary  circuit,  the  actual  or  total 
magnetic  flux  interlinked  with  the  circuit,  3>0,  is  that  due  to  the 
impressed  voltage,  EQ,  minus  the  resistance  drop,  r0/o,  $'o  =  EQ  — 
r0  fa.  Of  this  magnetic  flux,  3>0,  a  part,  $'0,  passes  as  primary  leak- 
age flux  between  primary  and  secondary,  without  reaching  the 
secondary,  and  is  represented  by  the  primary  reactance  voltage, 
jxofo,  and  the  remainder  —  usually  the  major  part  —  is  impressed 
upon  the  secondary  circuit  as  mutual  magnetic  flux,  <f>  =  <f>0  —  <£>'(), 
corresponding  to  the  mutual  inductive  voltage,  ^  =  fl'o  —  jx0fQ. 
The  mutual  magnetic  flux,  <f>,  then  is  impressed  upon  the  second- 


230  ELECTRIC  CIRCUITS 

ary,  and  as  stated  above,  a  part  of  it,  the  secondary  leakage  flux, 
<£'i,  is  shunted  across  outside  of  the  secondary  circuit,  the  re- 
mainder, <£'  =  <i>  —  <f>'i,  passes  through  the  secondary  circuit  and 
corresponds  to  ri/i. 

118.  Applying  this  to  the  polyphase  induction  motor  with 
single  squirrel-cage  secondary.     Let 

YQ  =  g   —  jb   =  primary  exciting  admittance; 

Zo  =  7*0  +  jx0  =  primary  self  -inductive  impedance; 

Zi  =  n  +  j%i  =  secondary  self-inductive  impedance 

at  full  frequency,  reduced  to  the  primary. 
Let 

$1  =  the  true  induced  voltage  in  the  secondary,  at  full 

frequency,  corresponding  to  the  magnetic  flux  in 

the  armature  core. 

The  secondary  current  then  is 

'-*•      :,         : 

The  mutual  inductive  voltage  at  full  frequency, 
E  =  Ei  +  j 


Thus  the  exciting  current, 

/oo  =  Y0E 


where 


gx 

and  the  total  current, 

7o  =  /i  +  /oo 

f  s 
=  El  I  -  +  qi  -  jqz 

hence,  the  primary  impressed  voltage, 


l  +  j         +  (r0  +  jxo)         +  ffi  - 
ci  +  jc2), 


REACTANCE  OF  INDUCTION  APPARATUS         231 

where 

Ci  =  1  H-  r0  (  --  h  qi)  +  XoQz  =  1  +  s  —  +  r0qi  + 

\7'  /  7* 


7'i  /  7*1 

s#i  /  s  \  s(xi  -f-  #o)    , 

c2  =  —  +  XQ{-  +  qi)   -  r0qz  =       —  ---  h  Xoqi  -  r0q2, 

choosing  now  the  impressed  voltage  as  zero  vector, 

#o  =  e0 
gives 

F  e° 

^  =  ^T7^2' 

or,  absolute, 


the  torque  of  the  motor  is 

D  =  /Ei,  7i/  1 


the  power, 


~          S6i2(l   —  s) 
P  =  L 


+  c22) 
the  volt-ampere  input, 

Q  =  e0io 
etc. 

As  seen,  this  method  is  if  anything,  rather  less  convenient 
than  the  conventional  method,  which  starts  with  the  mutual 
inductive  voltage  E. 

It  becomes  materially  more  advantageous,  however,  when 
dealing  with  double  and  triple  squirrel-cage  structures,  as  it 
permits  starting  with  the  innermost  squirrel-cage,  and  gradually 
building  up  toward  the  primary  circuit.  See  "Multiple  Squirrel- 
cage  Induction  Motor,"  "Theory  and  Calculation  of  Electrical 
Apparatus." 


CHAPTER  XIII 
REACTANCE  OF  SYNCHRONOUS  MACHINES 

119.  The  synchronous  machine — alternating-current  generator, 
synchronous  motor  or  synchronous  condenser — consists  of  an 
armature  containing  one  or  more  electric  circuits  traversed  by 
alternating  currents  and  synchronously  revolving  relative  to  a 
unidirectional  magnetic  field,  excited  by  direct  current.  The 
armature  circuit,  like  every  electric  circuit,  has  a  resistance,  r, 
in  which  power  is  being  dissipated  by  the  current,  7,  and  an  in- 
ductance, L,  or  reactance,  x  =  2  ir/L,  which  represents  the  mag- 
netic flux  produced  by  the  current  in  the  armature  circuit,  and 
interlinked  with  this  circuit.  Thus,  if  EQ  =  voltage  induced  in 
the  armature  circuit  by  its  rotation  through  the  magnetic  field — 
or,  as  now  more  usually  the  case,  the  rotation  of  the  magnetic 
field  through  the  armature  circuit — the  terminal  voltage  of  the 
armature  circuit  is 

$  =  E0-  (r+jx)f. 

In  Fig.  110  is  shown  diagrammatically  the  path  of  the  field  flux, 
in  two  different  positions,  A  with  an  armature  slot  standing  mid- 
way between  two  field  poles,  B  with  an  armature  slot  standing 
opposite  the  field  pole. 

In  Fig.  Ill  is  shown  diagrammatically  the  magnetic  flux  of 
armature  reactance,  that  is,  the  magnetic  flux  produced  by  the 
current  in  the  armature  circuit,  and  interlinked  with  this  circuit, 
which  is  represented  by  the  reactance  x,  for  the  same  two  relative 
positions  of  field  and  armature. 

As  seen,  field  flux  and  armature  flux  pass  through  the  same  iron 
structures,  thus  can  not  have  an  independent  existence,  but  actual 
is  only  their  resultant.  This  resultant  flux  of  armature  self-in- 
duction and  field  excitation  is  shown  in  Fig.  112,  for  the  same  two 
positions,  A  and  B,  derived  by  superpositions  of  the  fluxes  in  Figs. 
110  and  111. 

As  seen,  in  Fig.  112A,  all  the  lines  of  magnetic  forces  are  inter- 
linked with  the  field  circuit,  but  there  is  no  line  of  magnetic  flux 
interlinked  with  the  armature  circuit  only,  that  is,  there  is  ap- 

232 


REACTANCE  OF  SYNCHRONOUS  MACHINES      233 

parently  no  self-inductive  armature  flux,  and  no  true  self-induct- 
ive reactance,  x,  and  the  self-inductive  armature  flux  of  Fig.  Ill 
thus  merely  is  a  mathematical  fiction,  a  theoretical  component  of 
the  resultant  flux,  Fig.  112.  The  effect  of  the  armature  current, 


ARMATURE 


FIELD 


ARMATURE 


FIELD 


FIG.    110. 


in  changing  flux  distribution,  Fig.  110A  to  Fig.  112A,  consists  in 
reducing  the  field  flux,  that  is,  flux  in  the  field  core,  increasing 
the  leakage  flux  of  the  field,  that  is,  the  flux  which  leaks  from  field 
pole  to  field  pole,  without  interlinking  the  armature  circuit,  and 


234 


ELECTRIC  CIRCUITS 


still  further  decreasing  the  armature  flux,  that  is,  the  flux  issuing 
from  the  field  and  interlinking  with  the  armature  circuit. 

In  position  1125,  there  is  no  self-inductive  armature  flux  either, 
but  every  line  of  force,  which  interlinks  with  the  armature  circuit, 


ARMATURE 


FIELD 


ARMATURE 


FIELD 


FIG.  111. 

is  produced  by  and  interlinked  with  the  field  circuit.  The  effect 
of  the  armature  current  in  this  case  is  to  increase  the  field  flux  and 
the  flux  entering  the  armature  at  one  side  of  the  pole,  and  decrease 
it  on  the  other  side  of  the  pole,  without  changing  the  total  field 
flux  and  the  leakage  flux  of  the  field.  Indirectly,  a  reduction  of 


REACTANCE  OF  SYNCHRONOUS  MACHINES      235 

the  field  flux  usually  occurs,  by  magnetic  saturation  limiting  the 
increase  of  flux  at  the  strengthened  pole  corner;  but  this  is  a  sec- 
ondary effect. 


ARMATURE 


FIELD 


ARMATURE 


FIELD 


FIG.  112. 

As  seen,  in  112A  the  armature  current  acts  demagnetizing,  in 
1125  distorting  on  the  field  flux,  and  in  the  intermediary  position 
between  A  and  B,  a  combination  of  demagnetization  (or  magneti- 
zation, in  some  positions)  and  distortion  occurs. 

Thus,  it  may  be  said  that  the  armature  reactance  has  no  inde- 
pendent existence,  is  not  due  to  a  flux  produced  by  and  interlinked 


23G  ELECTRIC  CIRCUITS 

only  with  the  armature  circuit,  but  it  is  the  electrical  representa- 
tion of  the  effect  exerted  on  the  field  flux  by  them.m.f.  of  the  arma- 
ture current. 

Considering  the  magnetic  disposition,  an  armature  current, 
which  alone  would  produce  the  flux,  Fig.  Ill,  in  the  presence  of  a 
field  excitation  which  alone  would  give  the  flux,  Fig.  110,  has  the 
following  effect:  in  Fig.  Ill  A,  by  the  counter  m.m.f.  of  the  arma- 
ture current  the  resultant  m.m.f.  and  with  it  the  resultant  flux  are 
reduced  from  that  due  to  the  m.m.f.  of  field  excitation,  to  that 
due  to  field  excitation  minus  the  m.m.f.  of  the  armature  current. 
The  difference  of  the  magnetic  potential  between  the  field  poles  is 
increased :  in  Fig.  1 10A  it  is  the  sum  of  the  m.m.fs.  of  the  two  air- 
gaps  traversed  by  the  flux  (plus  the  m.m.f.  consumed  in  the  arma- 
ture iron,  which  may  be  neglected  as  small) ;  in  Fig.  112A  it  is  the 
sum  of  the  m.m.fs.  of  the  two  air-gaps  traversed  by  the  flux 
(which  is  slightly  smaller  than  in  Fig.  110A,  due  to  the  reduced 
flux)  plus  the  counter  m.m.f.  of  the  armature.  The  increased 
magnetic  potential  difference  causes  an  increased  magnetic  leak- 
age flux  between  the  field  poles,  and  thereby  still  further  reduces 
the  armature  flux  and  the  voltage  induced  by  it. 

In  Fig.  112J5,  the  m.m.f.  of  the  armature  current  adds  itself  to 
the  m.m.f.  of  field  excitation  on  one  side,  and  thereby  increases  the 
flux,  and  it  subtracts  on  the  other  side  and  decreases  the  flux,  and 
thereby  causes  an  unsymmetrical  flux  distribution,  that  is,  a  field 
distortion. 

120.  Both  representations  of  the  effect  of  armature  current  are 
used,  that  by  a  nominal  magnetic  flux,  Fig.  Ill ,  which  gives  rise 
to  a  nominal  reactance,  the  "synchronous  reactance  of  the  arma- 
ture circuit,"  and  that  by  considering  the  direct  magnetizing 
action  of  the  armature  current,  as  "  armature  reaction,"  and 
both  have  their  advantages  and  disadvantages. 

The  introduction  of  a  synchronous  reactance,  XQ,  and  correspond- 
ing thereto  of  a  nominal  induced  e.m.f.,  e0,  is  most  convenient  in 
electrical  calculations,  but  it  must  be  kept  in  mind,  that  neither 
eQ  nor  XQ  have  any  actual  existence,  correspond  to  actual  magnetic 
fluxes,  and  for  instance,  when  calculating  efficiency  and  losses,  the 
core  loss  of  the  machine  does  not  correspond  to  eQ,  but  corresponds 
to  the  actual  or  resultant  magnetic  flux,  Fig.  112.  Also,  in  deal- 
ing with  transients  involving  the  dissipation  of  the  magnetic 
energy  stored  in  the  machine,  the  magnetic  energy  of  the  result- 
ant field,  Fig.  112,  comes  into  consideration,  and  not  the — much 


REACTANCE  OF  SYNCPIRONOUS  MACHINES      237 

larger — energy,  which  the  fields  corresponding  to  e0  and  x0  would 
have.  Thus  the  short-circuit  transient  of  a  heavily  loaded  ma- 
chine is  essentially  the  same  as  that  of  the  same  machine  at  no- 
load,  with  the  same  terminal  voltage,  although  in  the  former  the 
field  excitation  and  the  nominal  induced  voltage  may  be  very 
much  larger. 

The  use  of  the  term  armature  reaction  in  dealing  with  the  effect 
of  load  on  the  synchronous  machine  is  usually  more  convenient 
and  useful  in  design  of  the  machine,  but  less  so  in  the  calculation 
dealing  with  the  machine  as  part  of  an  electric  circuit. 

Either  has  the  disadvantage  that  its  terms,  synchronous  react- 
ance or  armature  reaction,  are  not  homogeneous,  as  the  different 
parts  of  the  reactance  field,  Fig.  Ill,  which  make  up  the  difference 
between  Fig.  112  and  Fig.  110,  are  very  different  in  their  action, 
especially  in  their  behavior  at  sudden  changes  of  circuit  conditions. 

121.  Considering  the  magnetic  flux  of  the  armature  current, 
Fig.  111A,  which  is  represented  by  the  synchronous  reactance,  x0. 

A  part  of  this  magnetic  flux  (lines  a  in  Fig.  111A)  interlinks 
with  the  armature  circuit  only,  that  is,  is  true  self-inductive  or 
leakage  flux.  Another  part,  however,  (6)  interlinks  with  the 
field  also,  and  thus  is  mutual  inductive  flux  of  the  armature  cir- 
cuit on  the  field  circuit.  In  a  polyphase  machine,  the  resultant 
armature  flux,  that  is,  the  resultant  of  the  fluxes,  Fig.  Ill,  of  all 
phases,  revolves  synchronously  at  (approximately)  constant  in- 
tensity, as  a  rotating  field  of  armature  reaction,  and,  therefore,  is 
stationary  with  regard  to  the  synchronously  revolving  field,  F. 
Hence,  the  mutual  inductive  flux  of  the  armature  on  the  field, 
though  an  alternating  flux,  exerts  no  induction  on  the  field  circuit, 
is  indeed  a  unidirectional  or  constant  flux  with  regards  to  the 
field  circuit.  Therefore,  under  stationary  conditions  of  load,  no 
difference  exists  between  the  self-inductive  and  the  mutual  in- 
ductive flux  of  the  armature  circuit,  and  both  are  comprised  in  the 
synchronous  reactance,  x0.  If,  however,  the  armature  current 
changes,  as  by  an  increase  of  load,  then  with  increasing  armature 
current,  the  armature  flux,  a  and  6,  Fig.  Ill,  also  increases,  a, 
being  interlinked  with  the  armature  current  only,  increases  simul- 
taneously with  it,  that  is,  the  armature  current  can  not  increase 
without  simultaneously  increasing  its  self-inductive  flux,  a.  The 
mutual  inductive  flux,  b,  however,  interlinks  with  the  field  circuit, 
and  this  circuit  is  closed  through  the  exciter,  that  is,  is  a  closed 
secondary  circuit  with  regards  to  the  armature  circuit  as  primary, 


238  ELECTRIC  CIRCUITS 

and  the  change  of  flux,  b,  thus  induces  in  the  field  circuit  an  e.m.f . 
and  causes  a  current  which  retards  the  change  of  this  flux  com- 
ponent, b.  Or,  in  other  words,  an  increase  of  armature  current 
tends  to  increase  its  mutual  magnetic  flux,  6,  and  thereby  to  de- 
crease the  field  flux.  This  decrease  of  field  flux  induces  in  the  field 
circuit  an  e.m.f.,  which  adds  itself  to  the  voltage  impressed  upon 
the  field,  thereby  increases  the  field  current  and  maintains  the 
field  flux  against  the  demagnetizing  action  of  the  armature  cur- 
rent, causing  it  to  decrease  only  gradually.  Inversely,  a  decrease 
of  armature  current  gives  a  simultaneous  decrease  of  the  self- 
inductive  part  of  the  flux,  a  in  Fig.  Ill,  but  a  gradual  decrease  of 
the  mutual  inductive  part,  b,  and  corresponding  gradual  increase 
of  the  resultant  field  flux,  by  inducing  a  transient  voltage  in  the 
field,  in  opposition  to  the  exciter  voltage,  and  thereby  decreasing 
the  field  current. 

Every  sudden  increase  of  the  armature  current  thus  gives  an 
equal  sudden  drop  of  terminal  voltage  due  to  the  self-inductive 
flux,  a,  produced  by  it  (and  the  resistance  drop  in  the  armature 
circuit),  an  equally  sudden  increase  of  the  field  current,  and  then 
a  gradual  further  drop  of  the  terminal  voltage  by  the  gradual  ap- 
pearance of  the  mutual  flux,  b,  and  corresponding  gradual  decrease 
of  field  current  to  nominal.  The  reverse  is  the  case  at  a  sudden 
decrease  of  armature  current. 

The  extreme  case  hereof  is  found  in  the  momentary  short-cir- 
cuit currents  of  alternators,1  which  with  some  types  of  machines 
may  momentarily  equal  many  times  the  value  of  the  permanent 
short-circuit  current.  However,  this  phenomenon  is  not  limited 
to  short-circuit  conditions  only,  but  every  change  of  current  in 
an  alternator  causes  a  momentary  overshooting,  the  more  so,  the 
greater  and  more  sudden  the  change  is. 

122.  That  part  of  the  synchronous  reactance,  XQ,  which  is  due  to 
the  magnetic  lines,  a,  in  Fig.  Ill,  is  a  true  self-inductive  reactance, 
x,  and  is  instantaneous,  but  that  part  of  Xi  representing  the  flux 
lines,  6,  is  mutual  inductive  reactance  with  the  field  circuit,  x',  and 
is  not  instantaneous,  but  comes  into  play  gradually,  and  when- 
ever dealing  with  rapid  changes  of  circuit  conditions,  the  syn- 
chronous reactance,  XQ,  thus  must  be  divided  into  a  true  or  self- 
inductive  reactance,  x}  and  a  mutual  inductive  reactance,  x': 

X0    =   X  -r  X.' 

1See  "Theory  and  Calculation  of  Transient  Phenomena." 


REACTANCE  OF  SYNCHRONOUS  MACHINES      239 

The  change  of  the  flux  disposition,  caused  by  a  current  in  the 
armature  circuit,  from  that  of  Fig.  110  to  that  of  Fig.  112,  thus  is 
simultaneous  with  the  armature  current  and  instantaneous  with  a 
sudden  change  of  armature  current  only  as  far  as  it  does  not  in- 
volve any  change  of  the  flux  through  the  field  winding,  but  the 
change  of  the  flux  through  the  field  coils  is  only  gradual.  Thus 
the  flux  change  in  the  armature  core  can  be  instantaneous,  but 
that  in  the  field  is  gradual. 

This  difference  between  self-inductive  and  mutual  inductive 
reactance,  or  between  instantaneous  and  gradual  flux  change, 
comes  into  consideration  only  in  transients,  and  then  very  fre- 
quently the  instantaneous  or  self-inductive  effect  is  represented 
by  a  self-inductive  reactance,  x,  the  gradual  or  mutual  inductive 
effect  by  an  armature  reaction. 

The  relation  between  self-inductive  component,  x,  and  mutual 
inductive  component,  x',  varies  from  about  2  -r-  1  in  the  unitooth- 
high  frequency  alternators  of  old,  to  about  1  -f-  20  in  some  of  the 
earlier  turbo-alternators. 

In  those  synchronous  machines,  which  contain  a  squirrel-cage 
induction-motor  winding  in  the  field  faces,  for  starting  as  motors, 
or  as  protection  against  hunting,  or  to  equalize  the  armature 
reaction  in  single-phase  machines,  all  the  armature  reactance  flux, 
which  interlinks  with  the  squirrel-cage  conductors  (as  the  flux,  c, 
in  Fig.  1115),  also  is  mutual  inductive  flux,  and  such  machines 
thus  have  a  higher  ratio  of  mutual  inductive  to  self-inductive 
armature  reactance,  that  is,  show  a  greater  overshooting  of  cur- 
rent at  sudden  changing  of  load,  and  larger  momentary  short- 
circuit  currents. 

The  mutual  flux  of  armature  reactance  induces  in  the  field  cir- 
cuit only  under  transient  conditions,  but  under  permanent  cir- 
cuit conditions  the  mutual  inductance  of  the  armature  on  the 
field  has  no  inducing  action,  but  is  merely  demagnetizing,  and  the 
distinction  between  self-inductive  and  mutual  inductive  react- 
ance thus  is  unnecessary,  and  both  combine  in  the  synchronous 
reactance.  In  this  respect,  the  synchronous  machine  differs 
from  the  transformer;  in  the  latter,  self-inductance  and  mutual 
inductance  are  always  distinct  in  their  action. 

123.  In  permanent  conditions  of  the  circuit,  the  armature  re- 
actance of  the  synchronous  machine  is  the  synchronous  react- 
ance, XQ  =  x  +  x' ;  at  the  instance  of  a  sudden  change  of  circuit 
conditions,  the  mutual  inductive  reactance,  x',  is  still  non-exist- 


240  ELECTRIC  CIRCUITS 

ing,  and  only  the  self-inductive  reactance,  x,  comes  into  play. 
Intermediate  between  the  instantaneous  effect  and  the  permanent 
conditions,  for  a  time  up  to  one  or  more  sec.,  the  effective  reactance 
changes,  from  x  to  XQ}  and  this  may  be  considered  as  a  transient 
reactance. 

During  this  period,  mutual  induction  between  armature  cir- 
cuit and  field  circuit  occurs,  and  the  phenomena  in  the  synchron- 
ous machine  thus  are  affected  by  the  constants  of  the  field  circuit 
outside  of  the  machine.  That  is,  resistance  and  inductance  of  the 
field  circuit  appear,  by  mutual  induction,  as  part  of  the  armature 
circuit  of  the  synchronous  machine,  just  as  resistance  and  react- 
ance of  the  secondary  circuit  of  a  transformer  appear,  trans- 
formed by  the  ratio  of  turns,  as  resistance  and  reactance  in  the  pri- 
mary, in  their  effect  on  the  primary  current  and  its  phase  relation. 

Thus  in  the  synchronous  machine,  a  high  non-inductive  re- 
sistance inserted  into  the  field  circuit  (with  an  increase  of  the 
exciter  voltage  to  give  the  same  field  current)  while  without 
effect  on  the  permanent  current  and  on  the  instantaneous  current 
in  the  moment  of  a  sudden  current  change,  reduces  the  duration 
of  the  transient  armature  current;  an  inductance  inserted  into 
the  field  circuit  lengthens  the  duration  of  the  transient  and  changes 
its  shape. 

The  duration  of  the  transient  reactance  of  the  synchronous 
machine  is  about  of  the  same  magnitude  as  the  period  of  hunting 
of  synchronous  machines — which  varies  from  a  fraction  of  a 
second  to  over  one  sec.  The  reactance,  which  limits  the  current 
fluctations  in  hunting  synchronous  machines,  thus  is  neither  the 
synchronous  reactance,  XQ,  nor  the  true  self-inductive  reactance,  x, 
but  is  an  intermediate  transient  reactance;  the  current  change  is 
sufficiently  slow  that  the  mutual  induction  between  synchronous 
machine  armature  and  field  has  already  come  into  play  and  the 
field  begun  to  follow,  but  is  too  rapid  for  the  complete  develop- 
ment of  the  synchronous  reactance. 

124.  In  the  polyphase  machine  on  balanced  load,  the  mutual 
inductive  component  of  the  armature  reactance  has  no  inductive 
effect  on  the  field,  as  its  resultant  is  unidirectional  with  regard 
to  the  field  flux.  In  the  single-phase  machine,  however  (or 
polyphase  machine  on  unbalanced  load),  such  inductive  effect 
exists,  as  a  permanent  pulsation  of  double  frequency.  The 
mutual  inductive  flux  of  the  armature  circuit  on  the  field  circuit 
is  alternating,  and  the  field  circuit,  revolving  synchronously 


REACTANCE  OF  SYNCHRONOUS  MACHINES      241 

through  this  alternating  flux,  thus  has  an  e.m.f.  of  double  fre- 
quency induced  in  it,  which  produces  a  double-frequency  current 
in  the  field  circuit,  superimposed  on  the  direct  current  from  the 
exciter.  The  field  flux  of  the  single-phase  alternator  (or  poly- 
phase alternator  at  unbalanced  load)  thus  pulsates  with  double 
frequency,  and,  by  being  carried  synchronously  through  the 
armature  circuits,  this  double-frequency  pulsation  of  flux  in- 
duces a  triple-frequency  harmonic  in  the  armature. 

Thus,  single-phase  alternators,  and  polyphase  alternators 
at  unbalanced  load,  contain  more  or  less  of  a  third  harmonic 
in  their  voltage  wave,  which  is  induced  by  the  double-frequency 
pulsation  of  the  field  flux,  resulting  from  the  pulsating  armature 
reaction,  or  mutual  armature  reactance,  x'. 

The  statement,  that  three-phase  alternators  contain  no  third 
harmonics  in  their  terminal  voltages,  since  such  harmonics  neu- 
tralize each  other,  is  correct  only  for  balanced  load,  but  at  un- 
balanced load,  three-phase  alternators  may  have  pronounced 
third  harmonics  in  their  terminal  voltage,  and  on  single-phase 
short-circuit,  the  not  short-circuited  phase  of  a  three-phase 
alternator  may  contain  a  third  harmonic  far  in  excess  of  the 
fundamental. 

125.  Let  in  a  F-connected  three-phase  synchronous  machine, 
the  magnetic  flux  per  field  pole  be  &Q.  If  this  flux  is  distributed 
sinusoidally  around  the  circumference  of  the  armature,  at  any 
time,  t,  represented  by  angle,  0  =  2  irftj  the  magnetic  flux  enclosed 
by  an  armature  turn  is 

$  =  $0  cos  4> 

when  counting  the  time  from  the  moment  of  maximum  flux. 
The  voltage  induced  in  an  armature  circuit  of  n  turns  then  is 

d$ 

e\  =  n  —j-  =  c$o  sm  <f> 

where 

c  =  2  irfn 

If,  however,  the  flux  distribution  around  the  armature  circum- 
ference is  not  sinusoidal,  it  nevertheless  can,  as  a  periodic  func- 
tion, be  expressed  by 


$  =  $0  [cos  0  +  a2  cos  2(0  —  a2)  +  ^3  cos  3(0  —  a3)  + 

a4  cos  4(0  —  «4)  +    .    .    .   ] 

and  the  voltage  induced  in   one  armature  conductor,   by  the 

16 


242  ELECTRIC  CIRCUITS 

synchronous  rotation  through  this  flux,  is 

d$ 

-j-j-  =  TT/$O  [sin  0  +  2  a2  sin  2(0  -  a2)  -f  3  a3  sin  3(0-a-3)  + 

4  a4  sin  4(0  —  01!)  +    .    .    .  ] 

hence,  the  voltage  induced  in  one  full-pitch  armature  turn,  or  in 
two  armature  conductors  displaced  from  each  other  on  the  arma- 
ture surface  by  one  pole  pitch  or  an  odd  multiple  thereof, 

e  =  2  7T/$0[sin  0-J-3  a3  sin  3(0  — a3)  +5  a5  sin  5(0  —  a5)+    .    .    .   ] 

that  is,  the  even  harmonics  cancel. 

The  voltage  induced  in  one  armature  circuit  of  n  effective  series 
turns  then  is 

ei  =  c$0  [sin  0  +  63  sin  3(0  —  «3)  +  65  sin  5(0  —  a5)  +    •    •    •   ] 

where 

6a  —  3  a3,  65  =  5  ab)  etc.,  if  all  the  n  turns  are  massed  together, 
and  are  less,  if  the  armature  turns  are  distributed,  due  to  the 
overlapping  of  the  harmonics,  and  partial  cancellation  caused 
thereby.  As  known,  by  causing  proper  pitch  of  the  turn,  or 
proper  pitch  of  the  arc  covered  by  any  phase,  any  harmonic  can 
be  entirely  eliminated. 

The  second  and  third  phase  of  the  three-phase  machine  then 
would  have  the  voltage, 

e-2  =  c$0  [sin  (0  -  120°)  +  63-sin  3(0  -  «8  -  120°)  -f 

65  sin  5(0  -  0:5  -  120°)  +  •  .  .  ] 

=  c$0  [sin  (0  -  120°)  +  63  sin  3(0  -  «3)  +  bb  sin 

(5[0  -  aj  +  120°)  +   .  .  .] 

e3  =  c$0  [sin  (0  -  240°)  +  63  sin  3(0  -  «3)  + 

&5  sin  (5[0  -  aj  +  240°)]  +    .    .    .] 

As  seen,  the  third  harmonics  are  all  three  in  phase  with  each 
other;  the  fifth  harmonics  are  in  three-phase  relation,  but  with 
backward  rotation;  the  seventh  harmonics  are  again  in  three- 
phase  relation,  like  the  fundamentals,  the  ninth  harmonics  in 
phase,  etc. 

The  terminal  voltages  of  the  machine  then  are 

EI  =  es  —  ez  =  V3  c$0  [cos  0  —  65  cos  5(0  —  a5)  + 

67  cos  7  (0  —  0-7) h  •    •    •] 

and  corresponding  thereto  Ez  =  e\  —  e3  and  Es  =  ez  —  e\,  differ- 
ing from  Ei  merely  by  substituting  0  —  120°  and  0  —  240°  for  0. 


REACTANCE  OF  SYNCHRONOUS  MACHINES      243 

As  seen,  the  third  harmonic  eliminates  in  the  terminal  voltages 
of  the  three-phase  machine,  regardless  of  the  flux  distribution, 
provided  that  the  flux  is  constant  in  intensity,  that  is,  the  load 
conditions  balanced. 

126.  Assuming,  however,  that  the  load  on  the  three-phase 
machine  is  unbalanced,  causing  a  double-frequency  pulsation 
of  the  magnetic  flux, 

3>o  (1  +  «•  cos  2  0), 

assuming  for  simplicity  sinusoidal  distribution  of  magnetic  flux. 
The  flux  interlinked  with  a  full- pitch  armature  turn  then  is 

<J>  =  $0(1  +  a  cos  2  0)  cos  (0  —  a) 

=  $0  [cos  (</>  -  a)  +  ^  cos  (4  +'a>  +  g  cos  (30-  a)  J 

and  the  voltage  induced  in  an  armature  circuit  of  n  effective  turns, 
ei  =  n-^-  =  c$0  ^T [cos  (0-  a)  +  ^cos(0  +  a)  +^c 
=  c$0[sin  (0  —  a)  -f  £  sin  (<£  +  «)  +  -7^ sin  (3  </>  -  a 

or,  if  the  magnetic  flux  maximum  coincides  with  the  voltage 
maximum  of  the  first  phase,  a  =  0, 

ei  =  c$0[  (l  +|)  sin  <j>  +  -^  sin  3  <£J . 

In  the  second  phase,  the  flux  is  the  same,  ^>0  (1  +  a  cos  2  0), 
but  the  flux  interlinkage  120°  later,  thus, 

$  =  $0  (1  -f  a  cos  2  </>)  cos  (0  -  a  --  120°), 

and  the  voltage  of  the  second  phase  thus  is  derived  from  that  of 
the  first  phase,  by  substituting  a  +  120°  for  a, 

ez  =  c<I>o  [sin  (</>  -  a  -  120°)  +  |  sin  (0  +  a  +  120°)  + 

^  sin  (3  </>  -  a  -  120°)] 
and  the  third  phase, 

63  =  c$0  [sin  (0  -  a  -  240°)  +  ^  sin  (0  +  a  +  240°)  + 

y  sin  (3  </>-«-  240°)] 


244  ELECTRIC  CIRCUITS 

the  terminal  voltages  thus  are, 

Ei  =  63  —  62  =  V3  c$0  I  cos  (<f>  —  a)  —  I  cos  (</>  +  a)  + 

-^  cos  (3  0  -  a)] 
and  in  the  same  manner,  the  other  two  phases, 

#2  =  \/3  c$0  [cos  (0  -  a  -  120°)  -  ^  cos  (0  +  a  -f  120°)  - 

^  cos  (3  0  -  a  -  120°)] 
#3  =  "N/3  c$0  [cos  (<t>  -  a  -  240°)  -  ^  cos  (0  +  a  +  240°)  - 

!%os(30-a-240°)]. 
For  a  =  0,  this  gives 

E,  =  \/3  c$0[  (l  -  I)  cos  </>  -f  ^  cos  3  0J 

^2  =  V3  c$0  [  (l  -  ^)  cos  (</>  -  120°)  -  ?|  cos  (3  0  -  120°)] 

E3  =  V3  c$0[  (l  -  ^)  cos  (0  -  240°)  -  ^  cos  (3  0  -  240°)]. 


As  seen,  all  three  phases  have  pronounced  third  harmonics, 
and  the  third  harmonic  of  the  loaded  phase,  EI,  is  opposite  to 
that  of  the  unloaded  phases. 

If  a  =  1,  which  corresponds  about  to  short-circuit  conditions, 
as  it  makes  the  minimum  value  of  $o  equal  zero,  then  the  quadra- 
ture phase  of  the  short-circuited  phase,  E\,  becomes 

e\  =      0  °(sin  <£  -f  sin  3  <£), 

<B 

that  is,  the  third  harmonic  becomes  as  large  as  the  fundamental. 
Thus,  on  unbalanced  load,  such  as  on  single-phase  short-circuit, 
triple  harmonics  appear  in  the  terminal  voltages  of  a  three-phase 
generator,  though  at  balanced  loads  the  three-phase  terminal 
voltage  can  contain  no  third  harmonics. 


SECTION  III 

CHAPTER  XIV 

CONSTANT-POTENTIAL   CONSTANT-CURRENT    TRANS- 

FORMATION 

127.  The  generation  of  alternating-current  electric  power  prac- 
tically always  takes  place  at  constant  voltage.     For  some  pur- 
poses, however,  as  for  operating  series  arc  circuits,  and  to  a  lim- 
ited extent  also  for  electric  furnaces,  a  constant,  or  approximately 
constant  alternating  current  is  required.     While  constant  alter- 
nating-current arcs  have  largely  come  out  of  use  and  their  place 
taken  by  constant  direct-current  luminous  arc  circuits,  or  incan- 
descent lamps,  the  constant  direct  current  is  usually  derived  by 
rectification  of  constant  alternating-current  supply  circuits. 

Such  constant  alternating  currents  are  usually  produced  from 
constant- voltage  supply  circuits  by  means  of  constant  or  variable 
inductive  reactances,  and  may  be  produced  by  the  combination  of 
inductive  and  condensive  reactances;  and  the  investigation  of 
different  methods  of  producing  constant  alternating  current  from 
constant  alternating  voltage,  or  inversely,  constitutes  a  good 
application  of  the  terms  "impedance,"  admittance, "  etc.,  and 
offers  a  large  number  of  problems  or  examples  for  the  symbolic 
method  of  dealing  with  alternating-current  phenomena. 

Even  outside  of  arc  lighting,  such  combinations  of  inductance 
and  capacity  which  tend  toward  constant-voltage  constant-cur- 
rent transformation  are  of  considerable  importance  as  a  possible 
source  of  danger  to  the  system.  In  a  constant-current  circuit, 
the  load  is  taken  off  by  short-circuiting,  while  open-circuiting 
causes  the  voltage  to  rise  to  the  maximum  value  permitted  by 
the  power  of  the  generating  source.  Hence,  where  the  circuit 
constants,  with  a  constant-voltage  supply  source,  are  such  as  to 
approach  constant-voltage  constant-current  transformation,  as  is 
for  instance  the  case  in  very  long  transmission  lines,  open-circuit- 
ing may  lead  to  dangerous  or  even  destructive  voltage  rise. 

128.  With  an  inductive  reactance  inserted  in  series  to  an  alter- 

245 


246  ELECTRIC  CIRCUITS 

nating-current  non-inductive  circuit,  at  constant-supply  voltage, 
the  current  in  this  circuit  is  approximately  constant,  as  long  as  the 
resistance  of  the  circuit  is  small  compared  with  the  series  inductive 
reactance. 

Let 

$o  =  eo  =  constant  impressed  alternating  voltage; 
r  =  resistance  of  non-inductive  receiver  circuit; 

x0  =  inductive  reactance  inserted  in  series  with  this  circuit. 

The  impedance  of  this  circuit  then  is 

Z  =  r  +  jx0, 
and,  absolute, 

z  =  Vr2  +  XQ\ 
and  thus  the  current, 


Z       r  +  jx0 
and  the  absolute  value  is 

i  =  -°  =  -r^~=  (2) 

z        Vr2   +  *o2 

the  phase  angle  of  the  supply  circuit  is  given  by 

tan  0Q  =  -  (3) 


and  the  power  factor, 

If  in  this  case,  r  is  small  compared  with  XQ,  it  is 


cos  00  =  -*  (4) 


or,  expanded  by  the  binomial  theorem, 

J_.f1+/n'}-H  =  i-  £i  + 


\XQ/ 

hence, 


(5) 


that  is,  for  small  values  of  r,  the  current,  i,  is  approximately 
constant,  and  is 


CONSTANT-CURRENT  TRANSFORMATION       247 
For  small  values  of  r,  the  power-factor 

cos  6  =  - 
z 

is  very  low,  however. 

Allowing  a  variation  of  current  of  10  per  cent,  from  short- 
circuit  or  no-load,  r  =  0,  to  full-load,  or  r  =  n,  it  is,  substituted 
in  (2): 

No-load  current: 


XQ 


5-25- 


\ 


\ 


FIG.  113. 


Full-load  current: 


i  = 


=  0.9  t'o. 


Hence, 


=  0.9^, 


100 


so 


-50 


-10 


and  therefore, 

n  =  0.485  x0, 

and  the  power-factor,  from  (4),  is  0.437. 

That  is,  even  allowing  as  large  a  variation  of  current,  i,  as 
10  per  cent.,  the  maximum  power-factor  only  reaches  43.7  per 
cent.,  when  producing  constant-current  regulation  by  series 
inductance  reactance. 


248  ELECTRIC  CIRCUITS 

As  illustrations  are  shown,  in  Fig.  113,  for  the  constants: 

eo   =  6600  volts  applied  e.muf . ; 
XQ  =  792  ohms  series  reactance; 
the  current: 

6600 


+  7922 
8.33 

»-*(»» 


amp.; 


and  the  power-factor: 
cos  0  =  — 


792 

with  the  voltage  at  the  secondary  terminals: 

e  =  ri 
as  abscissas. 

129.  If  the  receiver  circuit  is  inductive,  that  is,  contains,  in 
addition  to  the  resistance,  r,  an  inductive  reactance,  x}  and  if 
this  reactance  is  proportional  to  the  resistance, 

x  =  kr, 

as  is  commonly  the  case  in  arc  circuits,  due  to  the  inductive 
reactance  of  the  regulating  mechanism  of  the  arc  lamp  (the 
effective  resistance,  r,  and  the  inductive  reactance,  x,  in  this 
case  are  both  proportional  to  the  number  of  lamps,  hence  pro- 
portional to  each  other),  it  is: 
total  impedance : 

Z  =  r  +  j  (x»  +  x)  =  r  +  j  (x0  +  kr); 
or  the  absolute  value  is 


z  =  Vr2  +  (z0  +  xY  =  Vr*  +  (XQ  + 

thus,  the  current 

j  =  eo 

and  the  absolute  value  is 


kr)2       *o     f~    2  kr   ,    r2(l 


r^J      (8) 


CONSTANT-CURRENT  TRANSFORMATION       249 

and  the  power-factor: 

cos  0o  =  r-  =  T        ==•  (9) 

By  the  binomial  theorem,  it  is 


kr      r2(2-k2) 

A      „     2 


2  fcr       r2(l  +  fc2)  *o  4  x0 

1  H-  — 1 • — —^-L 

Hence,  the  current 

(10) 


.  .  . 

XQ  4  XQ2 

that  is,  the  expression  of  the  current,  i  (10),  contains  the  ratio* 
—  ,  in  the  first  power,  with  k  as  coefficient,  and  if  therefore  k 

XQ 

is  not  very  small,  that  is,  the  inductive  reactance,  x  =  kr,  a 
very  small  fraction  of  the  resistance,  r,  the  current,  i,  is  not 
even  approximately  constant,  but  begins  to  fall  off  immediately, 
even  at  small  values  of  r. 
Assuming,  for  instance, 

k  =  0.4. 

That  is,  the  inductive  reactance,  x,  of  the  receiver  circuit  equals 
40  per  cent,  of  its  resistance,  r,  and  the  power-factor  of  the 
receiver  circuit  accordingly  is 

cos  0  =  -—:  —  - 


1  +  k2 

—  93  per  cent.; 
it  is,  substituted  in  (8), 


As  illustrations  are  shown,  in  the  same  Fig.  113,  for  the  constants: 

eQ  =  6600  volts  supply  e.m.f . ; 
xo  =  792  ohms  series  reactance; 
the  current: 

8'33  amp. 


This  current  is  shown  by  dotted  line. 

In  this  case,  in  an  inductive  circuit,  the  current,  i,  has  decreased 


250  ELECTRIC  CIRCUITS 

by  10  per  cent,  below  the  no-load  or  short-circuit  value  of  8.33 
amp.  that  is,  has  fallen  to  7.5  amp.,  at  the  resistance  r  =  187  ohms, 
or  at  the  voltage  of  the  receiving  circuit, 


e  =  i  VrM-  x2  =  ri  V  1  +  k*  =  1.077  ri  =  1500  volts; 

while,  in  the  case  of  a  non-inductive  load,  the  current  has  fallen 
off  to  7.5  amp.,  or  by  10  per  cent,  at  the  resistance  r  =  395  ohms, 
or  at  the  voltage  of  the  receiving  circuit :  e  =  2950  volts. 

130.  As  seen,  a  moderate  constant-current  regulation  can 
be  produced  in  a  non-inductive  circuit,  by  a  constant  series 
inductive  reactance,  at  a  considerable  sacrifice,  however,  of  the 
power-factor,  while  in  an  inductive  receiver  circuit,  the  con- 
stant-current regulation  is  not  even  approximate. 

To  produce  constant  alternating  current,  from  a  constant- 
potential  supply,  by  a  series  inductive  reactance,  over  a  wide 
range  of  load  and  without  too  great  a 
sacrifice  of  power-factor,  therefore  re- 
quires a  variation  of  the  series  inductive 
reactance  with  the  load.  That  is,  with 
increasing  load,  or  increasing  resistance 
of  the  receiver  circuit,  the  series  induc- 
tive reactance  has  to  be  decreased,  so  as 
to  maintain  the  total  impedance  of  the 
F  circuit,  and  thereby  the  current,  constant. 

In  constant-current  apparatus,  as  trans- 
formers from  constant  potential  to  constant  current,  or  regula- 
tors, this  variation  of  series  inductive  reactance  with  the  load 
is  usually  accomplished  automatically  by  the  mechanical  motion 
caused  by  the  mechanical  force  exerted  by  the  magnetic  field  of 
the  current,  upon  the  conductor  in  which  the  current  exists. 

For  instance,  in  the  constant-current  transformer,  as  shown 
diagrammatically  in  Fig.  114,  the  secondary  coils,  S,  are  arranged 
so  that  they  can  move  away  from  the  primary  coils,  P,  or  in- 
versely. Primary  and  secondary  currents  are  proportional 
to  each  other,  as  in  any  transformer,  and  the  magnetic  field 
between  primary  and  secondary  coils,  or  the  magnetic  stray  field, 
in  which  the  secondary  coils  float,  is  proportional  to  either  current. 
The  magnetic  repulsion  between  primary  coils  and  secondary  coils 
is  proportional  to  the  current  (or  rather  its  ampere-turns),  and 
to  the  magnetic  stray  field,  hence  is  proportional  to  the  square 
of  the  current,  but  independent  of  the  voltage.  The  secondary 


CONSTANT-CURRENT  TRANSFORMATION       251 

coils,  S,  are  counter-balanced  by  a  weight,  W,  which  is  adjusted 
so  that  this  weight,  W,  plus  the  repulsive  thrust  between  second- 
ary coils,  Sj  and  primary  coils,  P  (which,  as  seen  above,  is  propor- 
tional to  the  square  of  the  current),  just  balances  the  weight  of 
the  secondary  coils.  Any  increase  of  secondary  current,  as,  for 
instance,  caused  by  short-circuiting  a  part  of  the  secondary  load, 
then  increases  the  repulsion  between  primary  and  secondary  coils, 
and  the  secondary  coils  move  away  from  the  primary;  hence  more 
of  the  magnetic  flux  produced  by  the  primary  coils  passes  between 
primary  and  secondary,  as  stray  field,  or  self-inductive  flux, 
less  passes  through  the  secondary  coils,  and  therefore  the  second- 
ary generated  voltage  decreases  with  the  separation  of  the  coils, 
and  also  thereby  the  secondary  current,  until  it  has  resumed  the 
same  value,  and  the  secondary  coil  is  again  at  rest,  its  weight 
balancing  counterweight  plus  repulsion. 

Inversely,  an  increase  of  load,  that  is,  of  secondary  impedance, 
decreases  the  secondary  current,  so  causes  the  secondary  coils 
to  move  nearer  the  primary,  and  to  receive  more  of  the  primary 
flux;  that  is,  generate  higher  voltage. 

In  this  manner,  by  the  mechanical  repulsion  caused  by  the  cur- 
rent, the  magnetic  stray  flux,  or,  in  other  words,  the  series  induct- 
ive reactance  of  the  constant-current  transformer,  varies  auto- 
matically between  a  maximum,  with  the  primary  and  secondary 
coils  at  their  maximum  distance  apart,  and  a  minimum  with  the 
coils  touching  each  other.  Obviously,  this  automatic  action  is 
independent  of  frequency,  impressed  voltage,  and  character  of 
load. 

If  the  two  coils  P  and  S  in  Fig.  114  are  wound  with  the  same 
number  of  turns  and  connected  in  series  with  each  other  and  with 
the  circuit,  Fig.  114  is  a  constant-current  regulator,  or  a  regulating 
reactance,  that  is,  a  reactance  which  varies  with  the  load  so  as  to 
maintain  constant  current.  If  P  is  primary  and  S  secondary 
circuit,  Fig.  114  is  a  constant-current  transformer. 

Assuming  then,  in  the  constant-current  transformer  or  regula- 
tor or  other  apparatus,  a  device  to  vary  the  series  inductive 
reactance  so  as  to  maintain  the  current  constant.     Let 
Bo  =  e0  =  constant  =  impressed  e.m.f., 
Z    =  r+jx, 

=  r  (1  -\-  jk)  the  impedance  of  the  load,  and  let 
x0   =  inductive    series    reactance,     as    the    self-inductive 
internal  reactance  of  the  constant-current  transformer. 


252  ELECTRIC  CIRCUITS 

The  current  in  the  circuit  then  is 

/=  e° 


r  +  j(x0  -f  x) 
or,  the  absolute  value, 


i  = 


and,  to  maintain  the  current,  i,  constant  (i  = 
then  requires 

or,  transposed, 


or,  for 

x  =  kr, 


that  is,  to  produce  perfectly  constant  current  by  means  of  a 
variable  series  inductive  reactance,  this  series  reactance  must  be 
varied  with  the  load  on  the  circuit,  according  to  equation  (11)  or 
(12). 
For  non-inductive  load,  or  x  =  0,  it  is 


the  maximum  load,  which  can  be  carried,  is  given  by 

XQ    =    0 

and  is 

z  =  VV2  +  x2  =  r  Vl  +  k2  =  ~  (14) 

^o 

As  seen  from  equation  (13),  the  decrease  of  inductive  reactance, 
x0,  required  to  maintain  constant  current  with  non-inductive 
load,  is  small  for  small  values  of  resistance,  r,  when  the  r2  under 
the  root  is  negligible.  With  inductive  load,  equation  (11),  the 
inductive  reactance,  XQ,  has  still  further  to  be  decreased  by  the 
inductive  reactance  of  the  load,  x. 

Substituting: 

e0 

XQQ    =   — 
^0 

as  the  value  of  the  series  inductive  reactance  at  no-load  or  short- 
circuit,  equations  (11),  (12),  (13)  assume  the  form: 


CONSTANT-CURRENT  TRANSFORMATION       253 
General  inductive  load: 


XQ  =  Vzoo2  -  r2  -  x,  (14) 


Inductive  load  of  —    =  k: 
r 


XQ  =  Vzoo2  —  r2  —  kr  (15) 

Non-inductive  load : 

Xo  =   Vzoo2  -  r2.  (16) 

131.  As  seen,  a  constant  series  inductive  reactance  gives  an 
approximately  constant-current  regulation  with  non-inductive 
load,  but  if  the  load  is  inductive  this  regulation  is  spoiled. 
Inversely  it  can  be  shown,  that  condensive  reactance,  that  is,  a 
source  of  leading  current  in  the  load,  improves  the  constant- 
current  regulation. 

With  a  non-inductive  load,  series  condensive  reactance  exerts 
the  same  effect  on  the  current  regulation  as  series  inductive  re- 
actance; the  equations  discussed  in  the  preceding  paragraphs  re- 
main the  same,  except  that  the  sign  of  XQ  is  reversed  and  the  cur- 
rent always  leading. 

With  series  condensive  reactance,  condensive  reactance  in  the 
load  spoils,  inductive  reactance  in  the  load  improves  the  constant- 
current  regulation. 

That  is,  in  general,  a  constant  series  reactance  gives  approxi- 
mately constant-current  regulation  in  a  non-inductive  circuit, 
and  with  a  reactive  load  this  regulation  is  impaired  if  the  react- 
ance of  the  load  is  of  the  same  sign  as  the  series  reactance,  and  the 
regulation  is  improved  if  the  reactance  of  the  load  is  of  opposite 
sign  as  the  series  reactance. 

Since  a  constant-current  load  is  usually  somewhat  inductive,  it 
follows  that  a  constant  series  condensive  reactance  gives  a  better 
constant-current  regulation,  in  the  average  case  of  a  some- 
what inductive  arc  circuit,  than  the  constant  series  inductive 
reactance. 
Let 

$o  =  e0  =  constant  =  impressed,  or  supply  voltage. 
Z  =  r  +  jx  =  impedance    of    the    load,    or    the    receiver 
circuit,  and 

x  =  kr, 
that  is, 

Z  =  rlk 


254  ELECTRIC  CIRCUITS 

or,  absolute, 

Let  now  a  constant  condensive  reactance  be  inserted  in  series  with 
this  circuit,  of  the  reactance,  —  xc,  then  the  total  impedance  of  the 
circuit  is 

Z'  =  r  -  j  (xc  -  kr).  (17) 

The  current  is 

/  =  Y^ TV  (18) 

•  M       /j  /  /v»  //•/¥•!  ^  ' 

'  JV*<!  K'J 

or,  the  absolute  value  is 

i  =  - — ,  °  :  f-[Q\ 

Vr2  +  (xc  -  kr)2 
the  phase  angle  is 


tan  00  =  -  ^r^-  (20) 

and  the  power-factor  is 

cos  6Q  =      .          '      (21) 

Vr2  +  (xc  -  kr)* 

for 

k  =  0, 

or  non-inductive  load,  equations  (19)  and  (21)  assume  the  form: 

{  =  — _ _  °         and  cos  Q  —  —  — 


that  is,  the  same  as  with  series  inductive  reactance. 

From  equation  (19)  it  follows,  that  with  increasing  current,  t, 
from  no-load: 

r  =  0,  hence  :«0  =  —  (22) 

xc 

the  current,  iQ)  first  increases,  reaches  a  maximum,  and  then 
decreases  again.  When  decreasing,  it  once  more  reaches  the 
value,  z'o,  for  the  resistance,  r*i,  of  the  load,  which  is  given  by 

•  e°  e° 


(23) 


(xe  - 
hence,  expanded, 


and  the  maximum  value  through  which  i  passes  between  r  —  0 
and  r  =  ri,  is  given  by 

£  =  o 

dr       U> 


CONSTANT-CURRENT  TRANSFORMATION       255 

or 

^{r2  +  (xc  -  fcr)2J   =  0  =  2r  -  2k(xc  -  kr); 

hence, 

kxc       _  TI  .     . 

_L       ~~|  /L  sU 

This  maximum  value  is  given  by  substituting  (24)  in  (19),  as 


for  =  i0  VI  +  fc2  (25) 

k  =  0.4, 
this  value  is 

i*  =  1.077  ?0> 

that  is,  the  current  rises  from  no-load  to  a  maximum  7.7  per  cent, 
above  the  no-load  value,  and  then  decreases  again. 
As  an  example,  let 

eQ  =  6600  volts  impressed  e.m.f. 
and 

xc  =  880  ohm  condensive  reactance, 
xc  being  chosen  so  as  to  give 

io  =  --  =  7.5  amp.; 

Xc 

for 

k  =  0.4, 
then, 

6600 

A      —    


COS  60   = 


(880-0.4r)2' 
r 


Vr2+(880-0.4r)2' 
e  =  zi  =  1.077  ri. 

These  values  of  current  and  power-factor  are  plotted,  with  the 
receiver  voltage  as  abscissae,  in  Fig.  115. 

132.  The  conclusions  from  the  preceding  are  that  a  constant 
series  reactance,  whether  condensive  or  inductive,  when  inserted 
in  a  constant-potential  circuit,  tends  toward  a  constant-current 
regulation,  at  least  within  a  certain  range  of  load.  That  is,  at 
varying  resistance,  r,  and  therefore  varying  load,  the  current  is 
approximately  constant  at  light  load,  and  drops  off  only  gradu- 
ally with  increasing  load. 


256 


ELECTRIC  CIRCUITS 


This  constant-current  regulation,  and  the  power-factor  of  the 
circuit,  are  best  if  the  reactance  of  the  receiver  circuit  is  of  oppo- 
site sign  to  the  series  reactance,  and  poorest  if  of  the  same  sign. 
That  is,  series  condensive  reactance  in  an  inductive  circuit,  and 
series  inductive  reactance  in  a  circuit  carrying  leading  current, 


% 
_100 

X 

90 

8 

i 

X 

x 

80 

7 

^ 

< 

70 

6 

x 

x 

\ 

N 

60 

to 

5S 

U.UJ 

X 

s 

\ 

50 

• 
< 
4 

4 

y 

\ 

40. 

3 

/ 

X" 

30 

? 

X 

20- 

I 

X 

/ 

/ 

10 

/ 

?r' 

'<. 

i. 

| 

KILOVOLTS 
1 

1 

e 

A 

FIG.  115. 

give  the  best  regulation;  series  inductive  reactance  with  an  in- 

ductive, and  series  condensive  reactance  with  leading  current  in 

the  circuit,  give  the  poorest  regulation. 

Since  the  receiver  circuit  is  usually  inductive,  to  get  best  regula- 

tion, either  a  series  condensive  reactance  has  to  be  used,  as  in  Fig. 

115,  or,  if  a  series  inductive  reactance 
is  used,  the  current  in  the  receiver  cir- 
cuit is  made  leading,  as,  for  instance, 
by  shunting  the  receiver  circuit  by  a 
condensive  reactance. 

Assuming,  then,  as  sketched  diagram- 
matically  in  Fig.  116,  in  a  circuit  of 

constant  impressed  e.m.f.,  E0  —  e0  =  constant,  a  constant  in- 

ductive reactance,  zo,  inserted  in  series;  and  the  receiver  circuit, 

of  impedance, 


1 


f 

i 


X0 


FIG  116. 


where 


=  r  +  jx  =  r(l  +  jk) 


tangent  of  the  angle  of  lag  =  — ; 


CONSTANT-CURRENT  TRANSFORMATION       257 

let  the  receiver  circuit  be  shunted  by  a  constant  condensive  react- 
ance, xc'  let  then: 

f]  =  potential  difference  of  receiver  circuit  or  the  condenser 
terminals, 

/  =  current  in  the  receiver  circuit,  or  the  "  secondary  current/' 

/i  =  current  in  the  condenser, 

/o  =  total  supply  current,  or  "primary  current." 

Then  /o  =  /  +  /i  (26) 

and  the  e.m.f.  at  receiver  circuit  is 

?  =  ZI  (27) 

at  the  condenser, 

f]=-jxcli  (28) 

hence, 

Ii-JjI  (29) 

AC 

and,  in  the  main  circuit,  the  impressed  e.m.f.  is 

E0  =  e0  =  #  +  jx/o  -  (30) 

Hence,  substituting  (26),  (27)  and  (29)  in  (30), 


eQ  =  ZI  +jxJl  +J-A 

\  X       I 


or 

e0  =  \ZXe     X°  +  jx0\I  (31) 

and 

/   =     x  _*°o  (32) 

Z  — —  — h  jxo 

If  xc  =  x0,  that  is,  if  the  shunted  condensive  reactance  equals 
the  series  inductive  reactance,  equations  (32)  assume  the  form, 

I  =  +^=-Jef  (33) 

J£Q  XQ 

and  the  absolute  value  is 

(34) 


that  is,  the  current,  i,  is  constant,  independent  of  the  load  and 
the  power-factor. 


17 


258  ELECTRIC  CIRCUITS 

That  is,  if  in  a  constant-potential  circuit,  of  impressed  e.m.f.,  eQ, 
an  inductive  reactance,  x0,  and  a  condensive  reactance,  xc,  are 
connected  in  series  with  each  other,  and  if 

xc  =  x0,  (35) 

that  is,  the  two  reactances  are  in  resonance  condition  with  each 
other,  any  circuit  shunting  the  capacity  reactance  is  a  constant- 
current  circuit,  and  regardless  of  the  impedance  of  this  circuit, 
Z  =  r  -\-  jXj  the  current  in  the  circuit  is 

,•  =  £  \ 

133.  Such  a  combination  of  two  equal  reactances  of  opposite 
sign  can  be  considered  as  a  transforming  device  from  constant 
potential  to  constant  current. 

Substituting,  therefore,  (35)  in  the  preceding  equation  gives: 
(33)  substituted  in  (29) : 

Current  in  shunted  capacity 

7i  =  j^2e0  (36) 

or,  absolute, 

il  =  ^  (37) 

and,  substituting  (33)  and  (36)  in  (26) : 
primary  supply  current  is 

/  o  =  5 —  €Q  (38) 

£o 

or  the  absolute  value  is 

/    2      i      7  V2"  f^Q^ 

and  the  power-factor  of  the  supply  current  is 

tan  60  = — ,     cos  00  =      .  (40) 

In  this  case,  the  higher  the  inductive  reactance,  x,  of  the 
receiving  circuit  the  lower  is  the  supply  current,  z'0,  at  the  same 
resistance,  r,  and  the  higher  is  the  power-factor,  and  if  x  =  XQ 

I0  =  —z  and  cos  0  =  1  (41) 

XQ2 

that  is,  the  primary,  or  supply  circuit  is  non-inductive,  and  the 
primary  current  is  in  phase  with  the  supply  e.m.f.,  and  the 


CONSTANT-CURRENT  TRANSFORMATION       259 


power-factor  is  unity,  while  the  secondary  or  receiver  current 
(33)  is  90°  in  phase  behind  the  primary  impressed  e.m.f.,  eQ. 

Inserting,  therefore,  an  inductive  reactance,  Xi  —  XQ  —  x,  in 
series  in  the  receiver  circuit  of  impedance,  Z  =  r  +  jx,  raises 
the  power-factor  of  the  supply  current,  iQ,  to  unity,  and  makes 
this  current,  t0,  a  minimum.  Or,  if  the  inductive  reactance, 
XQ,  is  inserted  in  the  receiver  circuit,  thus  giving  a  total  imped- 
ance, Z  +  jxQ  =  r  -f-  j  (x  +  XQ)  by  equation  (38),  substituting 
Z  +  jxQ  instead  of  Z,  gives  the  primary  supply  current  as 


T 

*° 


or  the  absolute  value  as 


ze0 

~2 

XQ 


(42) 
(43) 


FIG.  117. 


and  the  tangent  of  the  primary  phase  angle 

X 

tan  00  =  -  =  tan  0, 

that  is,  the  primary  power-factor  equals  that  of  the  secondary. 

Hence,  as  shown  diagrammatic- 
ally  in  Fig.  117,  a  combination 
of  two  equal  inductive  reactances 
in  series  with  each  other  and  with 
the  receiver  circuit,  and  shunted 
midway  between  the  inductive  re- 
actances by  a  condensive  reactance 
equal  to  the  inductive  reactance, 
transforms  constant  potential  into  constant  current,  and  inversely, 
without  any  change  of  power-factor,  that  is,  the  primary  supply 
current  has  the  same  power-factor  as  the  secondary  current. 

With  an  inductive  secondary  circuit,  the  primary  power- 
factor  can  in  this  case  be  made  unity,  by  reducing  the  inductive 
reactance  of  the  secondary  side,  by  the  amount  of  secondary 
reactance. 

134.  Shunted  condensive  reactance,  xc,  and  series  inductive 
reactance,  XQ,  therefore  transforms  from  constant  potential, 
60,  to  constant  current,  i,  and  inversely,  if  their  reactances  are 
equal,  xc  =  XQ,  and  in  this  case,  the  main  current  is  leading,  with 
non-inductive  load,  and  the  lead  of  the  main  current  decreases, 
with  increasing  inductive  reactance,  that  is,  increasing  lag,  of  the 


260 


ELECTRIC  CIRCUITS 


receiving  circuit.  The  constant  secondary  current,  i,  lags  90° 
behind  the  constant  primary  e.rn.f.,  e0. 

Inversely,  by  reversing  the  signs  of  x0  and  xc  in  the  preceding 
equations,  that  is,  exchanging  inductive  and  condensive  react- 
ances, it  follows  that  shunted  inductive  reactance,  XQ,  and 
series  condensive  reactance,  xc,  if  of  equal  reactance,  xc  =  XQ, 
transform  constant  potential,  e0,  into  constant  current,  i,  and 
inversely.  In  this  case,  the  main  current  lags  the  more  the 
higher  the  inductive  reactance  of  the  receiving  circuit,  and 
the  constant  secondary  current,  i,  is  90°  ahead  of  the  constant 
primary  e.m.f.,  e0. 

In  general,  it  follows  that,  if  equal  inductive  and  condensive 
reactances,  XQ  =  xc,  that  is,  in  resonance  conditions,  are  con- 
nected in  series  across  a  constant-potential  circuit  of  impressed 


IV 


e.m.f.,  eQ,  any  circuit  connected  to  the  common  point  between 
the   reactances   is   a   constant-current   circuit,   and   carries   the 

,    .       e0 
current,  i  =  — . 

XQ 

Instead  of  connecting  this  secondary  or  constant-current 
circuit  with  its  other  terminal  to  line,  A,  so  shunting  the  con- 
densive reactance  with  it,  and  causing  the  main  current  to  lead 
(I  in  Fig.  118),  or  to  line,  B,  so  shunting  the  inductive  reactance 
with  it,  and  causing  the  main  current  to  lag  (II  in  Fig.  118),  it 
can  be  connected  to  any  point  intermediate  between  A  and  B, 
by  a  autotransf  ormer  as  in  III,  Fig.  1 18.  If  connected  to  the  mid- 
dle point  between  A  and  B,  the  main  current  is  neither  lagging  nor 
leading,  that  is,  is  non-inductive,  with  non-inductive,  load,  and 
with  inductive  load,  has  the  same  power-factor  as  the  load. 

The  two  arrangements,  I  and  II,  can  also  be  combined,  by 
connecting  the  constant-current  circuit  across,  as  in  IV,  Fig.  118, 
and  in  this  case  the  two  inductive  reactances  and  two  conden- 


CONSTANT-CURRENT  TRANSFORMATION       261 

sive  reactances  diagrammatically  form  a  square,  with  the  con- 
stant potential,  e0)  as  one,  the  constant  current,  i,  as  the  other 
diagonal,  as  shown  in  Fig.  119.  This  arrangement  has  been 
called  the  monocyclic  square. 

The  insertion  of  an  e.m.f.  into  the  constant-current  circuit, 
in  such  arrangements,  obviously,  does  not  exert  any  effect  on 
the  constancy  of  the  secondary  current,  i,  but  merely  changes 
the  primary  current,  iQ,  by  the  amount  of  power  supplied  or 
consumed  by  the  e.m.f.  inserted  in  the  secondary  circuit. 

While  theoretically  the  secondary  current  is  absolutely  con- 
stant, at  constant  primary  e.m.f.,  practically  it  can  not  be  per- 
fectly constant,  due  to  the  power 
consumed  in  the  reactances,  but 
falls  off  slightly  with  increase  of 
load,  the  more,  the  greater  the 
loss  of  power  in  the  reactances, 
that  is,  the  lower  the  efficiency  of 
the  transforming  device. 

Two  typical  arrangements  of 
such  constant-current  transform- 
ing devices  are  the  T-connection 
or  the  resonating-circuit,  diagram 
Fig.  117,  and  the  monocyclic  "  FIQ 

square,  diagram  Fig.  119.  From 
these,  a  very  large  number  of  different  combinations  of  in- 
ductive and  condensive  reactances,  with  addition  of  autotrans- 
formers,  and  of  impressed  e.m.fs.,  can  be  devised  to  transform 
from  constant  potential  to  constant  current  and  inversely,  and 
by  the  use  of  quadrature  e.m.fs.  taken  from  a  second  phase  of 
the  polyphase  system,  the  secondary  output,  for  the  same  amount 
of  reactances,  increased. 

These  combinations  afford  very  convenient  and  instructive 
examples  for  accustoming  oneself  to  the  use  of  the  symbolic 
method  in  the  solution  of  alternating-current  problems. 

Only  two  typical  cases,  the  T-connection  and  the  monocyclic 
square  will  be  more  fully  discussed. 

A.  T-Connection  or  Resonating  Circuit 

135.  General. — A  combination,  in  a  constant-potential  circuit, 
flf  an  inductive  and  a  condensive  reactance  in  series  with  each 


262  ELECTRIC  CIRCUITS 

other  in  resonance  condition,  that  is,  with  the  condensive  react- 
ance equal  to  the  inductive  reactance,  gives  constant  current  in 
a  circuit  shunting  the  capacity.  This  circuit  thus  can  be  called 
the  "secondary  circuit"  of  the  constant  potential  constant- 
current  transforming  device,  while  the  constant-potential  supply 
circuit  may  be  called  the  "primary  circuit." 

If  the  total  inductive  reactance  in  the  constant-current  cir- 
cuit is  equal  to  the  condensive  reactance,  the  primary  supply 
current  is  in  phase  with  the  impressed  e.m.f. 

Let,  as  shown  diagrammatic  ally  in  Fig.  117, 
x0  =  value  of  the  inductive  and  the  condensive  reactances  which 

are  in  series  with  each  other. 

Xi  =  the  additional  inductive  reactance  inserted  in  the  constant- 
current  circuit. 

Z  =  r  +  jx,  or  z  =  VV2  +  #2  =  the  absolute  value  of  the  im- 
pedance of  the  constant-current  load. 

Assuming  now  in  the  constant-current  circuit  the  inductive 
reactance  and  the  resistance  as  proportional  to  each  other,  as 
for  instance  is  approximately  the  case  in  a  series  arc  circuit, 
in  which,  by  varying  the  number  of  lamps  and  therewith  the 
load,  reactance  and  resistance  change  proportionally.  Let,  then, 

/*» 

k  =  -  =  ratio  of  inductive  reactance  to  resistance  of  the  load, 

or  tangent  -of  the  angle  of  lag  of  the  constant-current  circuit. 
It  is  then 

Z  =  r(l-+jk) 

and  z=  rVl  +  k2  (1) 

let,  then, 

$o  =  eo  =  constant  =  primary  impressed  e.m.f.,  or  sup- 
ply voltage, 

$1  =  potential  difference  at  condenser  terminals, 
$    =  secondary   e.m.f.,    or   voltage  at  constant-current 

circuit, 

/o    =  primary  supply  current, 
/i     =  condenser  current, 
/      =  secondary  current, 

then,  in  the  secondary  or  receiver  circuit, 

$-Zl  (2) 

at  the  condenser  terminals 


CONSTANT-CURRENT  TRANSFORMATION       263 

#!    =     E    +   JW 

=  (Z+jxJI  (3) 

and,  also, 

#!  =   -  jxo/i  (4) 

hence, 

h  =  ,-  qLffi/  (5) 

and  the  primary  current  is 


hence,  expanded, 


and  the  primary  supply  voltage  is 


hence,  substituting  (3)  and  (6), 

r /  *7    \     *    \       f  'x 
or,  expanded, 

60    =+./*»/  (7) 

or,  the  secondary  current  is 

T  J^O  /Q\ 

I  =   ~-  —  (o) 

and,  substituting  (8)  in  (6)  and  (5) :  the  primary  current  is 

/  0    ==    ~~  o  ^0  \"/ 

the  condenser  current  is 


or,  the  absolute  value  is 


XQ 

(ii) 

(12) 
(13) 

/-I     A  \ 

It     —                                                                fn 

•^                  7             •                      .  1                                          i                           1                                        I 

tan  0  =  -  =  k  gives  the  secondary  phase  angle  (14) 

and 

tan  00  = —  gives  the  primary  phase  angle  (15) 


264  ELECTRIC  CIRCUITS 

This  phase  angle  0i  =  0,  that  is,  the  primary  supply  current  is 
non-inductive,  if 

XQ  —  Xi  —  x  =  0, 
that  is, 

Xi   =   XQ   —X.  (16) 

The  primary  supply  can  in  this  way  be  made  non-inductive  for 
any  desired  value  of  secondary  load,  by  choosing  the  reactance,  x\, 
according  to  equation  (16). 

If  x  =  0,  that  is,  a  non-inductive  secondary  circuit  (series  in- 
candescent lamps  for  instance),  x\  =  XQ,  that  is,  with  a  non-in- 
ductive secondary  circuit,  the  primary  supply  current  is  always 
non-inductive,  if  the  secondary  reactance,  x\,  is  made  equal  to  the 
primary  reactance,  XQ. 

In   this    case   x\  =  XQ,    with   an   inductive   secondary  circuit 

/j* 

tan  d0  =  -  —  tan  6;  that  is,  the  primary  supply  current  has  the 

same  phase  angle  as  the  secondary  load,  if  all  three  reactances 
(two  inductive  and  one  condensive  reactance)  are  made  equal. 

In  general,  x\  would  probably  be  chosen  so  as  to  make  /o  non- 
inductive  at  full-load,  or  at  some  average  load. 

136.  Example. — A  100-lamp  arc  circuit  of  7.5  amp.  is  to  be 
operated  from  a  6600-volt  constant-potential  supply  eQ  =  6600 
volts,  and  i  —  7.5  amp. 

Assuming  75  volts  per  lamp,  including  line  resistance,  gives  a 
maximum  secondary  voltage,  for  100  lamps,  of  e'  =  7500  volts. 

Assuming  the  power-factor  of  the  arc  circuit  as  93  per  cent, 
lagging,  gives 

cos  6  -  0.93,  or  tan  0  =  0.4; 
hence, 

k  =  -  =  0.4,  and  Z  =  r(l  +  0.4j), 

or  z  =  1.077  r  at  full-load, 

if  ef  =  7500  volts, 

ef 
z'  =  -  =  1000  ohms, 

hence 

r'  =  0.93  z'  =  930  ohms, 

x'  =  0.4  r'  =  372  ohms, 
and 

e0  CQ       6600       oon    , 

i  =  — >  or  XQ  =  —  =  -=-=-  =  880  ohms. 

XQ  I  7.5 


CONSTANT-CURRENT  TRANSFORMATION       265 

To  make  the  primary  current  IQ  non-inductive  at  full-load,  or  for 
x'  =  372  ohms,  this  requires 

Xi  =  XQ  —  x'  =  508  ohms. 
This  gives  the  equations 

i  =  7.5  amp., 

e  =  7.52  =  8.08  r  volts. 

6600 


*o  =  Jr2-h  (372  -  0.4  r)2  X 


8802 


7-5 


+ 


880  -  2200 


hence,  leading  current  below  full-load,  non-inductive  at  full-load 
and  lagging  current  at  overload. 

137.  Apparatus  Economy.  —  Denoting  by  z',  r',  x'  the  respective 
full-load  values,  the  volt-ampere  output  at  full-load  is 


volt-ampere  input, 

Q  -  wo  -      -  (18) 


That  is,  the  volt-ampere  input  is  less  than  the  volt-ampere 
output,  since  the  input  is  non-inductive,  while  the  output  is  not. 
The  power  output  is 

p  =  w  =  ^  (19) 

which  is  equal  to  the  volt-ampere  input,  since  the  losses  of  power 
in  the  reactances  were  neglected  in  the  preceding  equations. 
The  volt-amperes  at  the  condenser  are 

Q'  =  *i2z0; 
hence,  substituting  (13), 

Q'   =  ^  +  (*'  +  *Q2     2  _  i"  +  (to*  +  s,)«  . 

3  (2°} 


The  volt-ampere  consumption  of  the  first,  or  primary  inductive 
reactance,  x0,  is 


266  ELECTRIC  CIRCUITS 

hence,  substituting  (12), 

r"  +  (XQ  -  x'  -  *,)'  .  ,  _  r"  +  (g.  -  kr'  -  ».)t 
^  ~^~  ~^~ 

the  volt-ampere  consumption  of  the  second,  or  secondary  induct- 
ive reactance,  Xi,  is 

Q"'  -  f*,f 

or 

Q'"--^«o»  (22) 

#0 

The  total  volt-ampere  rating  of  the  reactances  required  for  the 
transformation  from  constant  potential  to  constant  current  then  is 

Q  =  Q'  +  Q"  +  Q'" 

2  r'2(1  +  fc2)  +  2  kr'(2Xl  -  XQ)  +  (so2  -  o^  +  2^)     2 
~^7~ 

and  the  apparatus  economy,  or  the  ratio  of  volt-amperes  output 
to  the  volt-ampere  rating  of  the  apparatus  is 


,  =.      o^  =  _  r 

Q       2r/2(l  +  fc2)  +  2  fcr'(2  a*  -  x0)  +  (^o2  -  xoXl  +  2  a^i2) 

(24) 

this  apparatus  economy  depends  upon  the  load,  r',  the  power- 
factor  or  phase  angle  of  the  load,  k,  and  the  secondary  additional 
inductive  reactance,  x\. 

To  determine  the  effect  of  the  secondary  inductive  reactance, 
Xii  The  apparatus  economy  is  a  maximum  for  that  value  of 

secondary  inductive  reactance,  Xj,  for  which  -j—  =  0. 

Instead  of  directly  differentiating  /,  it  is  preferable  to  simplify 
the  function  /  first,  by  dropping  all  those  factors,  terms,  etc., 
which  inspection  shows  do  not  change  the  position  of  the  maxi- 
mum or  the  minimum  value  of  the  function.  Thus  the  numera- 
tor can  be  dropped,  the  denominator  made  numerator,  and  its 
first  term  dropped,  leaving 

/'  =  2  kr'  (2  X!  -  x0)  +  (*o2  -  x0xi  +  2  xj] 

as  the  simplest  function,  which  has  an  extreme  value  for  the  same 
value  of  xi,  as  /.  Then 

-^-'  =  4  kr'  -  x0  +  4  xi  =  0, 
ax  i 

.  x0  —  4  kr'  /oex 

and  xi  =  -  (25) 


CONSTANT-CURRENT  TRANSFORMATION       267 

substituting  (25)  in  (24),  gives 


f    =          8  r'xp     l  +  /26) 

"  16r'2-8/cr'x0 


To  determine  the  effect  of  the  load  r' : 

fi  becomes  a  maximum  for  that  load,  r',  which  makes 

<F  =  °' 
or,  simplified, 

,    =  16  r'2  -  8  kr'x0  +  7  x02 

J     1  „/  J 

hence 

^p  =  r'(32  r'  -  8  kx0)  -  (16  r'2  -  8  kr'x0  +  7  x02)  =  0, 

hence  r1  =  -  (27) 

and,  substituting  (27)  in  (26), 

/2  =   y  +  fc2  (28) 

hence,  for  fc  =  0: 

/2  =  v^  =  °'378' 

''7 

=  0.662  x0, 


°  =  0.25  x0, 


for  &  =  0.4: 


/2  =  _  0  478 

V7  -  0.4 
/       W7 


=  0.712 


Xl  »  5?  (1  _  0.4 \/7)  =  -  0.016  x0 

=  approximately  zero. 
At  non-inductive  load 

k  =0 
and  with  non-inductive  primary  supply,  that  is, 

Xi   =Xo, 


268  ELECTRIC  CIRCUITS 

by  substituting  these  values  in  (24),  the  apparatus  economy  is 

°  (29) 


2(r'2+zo2) 
which  is  a  maximum  for 

r'  =  XQ  (30) 

/o  =  \  =  0.25  (31) 

which  is  rather  low  : 

That  is,  non-inductive  load  and  supply  circuit  do  not  give  very 
high  apparatus  economy,  but  inductive  reactance  of  the  load,  and 
phase  displacement  in  the  supply  circuit,  gives  far  higher  appa- 
ratus economy,  that  is,  more  output  with  the  same  volt-amperes 
in  reactance. 

By  inserting  in  (23),  with  the  quantities,  Q',  Q",  and  Q"\ 
coefficients  n\t  nz,  ns,  which  are  proportional  respectively  to  the 
cost  of  the  reactances  per  kilovolt-ampere,  the  expression 

(32) 


then  represents  the  commercial  economy,  that  is,  the  maximum 
of  this  expression,  derived  by  analogous  considerations  as  before, 
gives  the  arrangement  for  minimum  cost  at  given  output. 

138.  Power  Losses  in  Reactances.  — 

In  the  preceding  equations,  the  losses  of  power  in  the  reactances 
have  been  neglected.  However  small  these  may  be,  in  accurate 
investigations,  they  require  consideration  as  to  their  effect  on  the 
regulation  of  the  transforming  device,  and  on  the  efficiency. 

Let 

a  =  power-factor  of  inductive  reactance,  that  is,  loss  of  power, 
as  fraction  of  total  volt-amperes. 

b  =  power-factor  of  condensive  reactance,  that  is,  loss  of  power, 
as  fraction  of  total  volt-amperes. 

Here  a  and  b  are  very  small  quantities,  in  general  6,  the  loss  in 
the  condensive  reactance,  being  far  smaller  than  the  loss  in  the 
inductive  reactance. 

Approximately,  the  inductive  reactances  are  (a  +  j)^o  and 
(a  +  j)xi  respectively,  and  the  condensive  reactance  is  (b  —j)x0. 

Assuming  the  same  denotations  as  in  the  preceding  paragraphs, 
receiver  circuit 

$  =  ZJ  (33) 


CONSTANT-CURRENT  TRANSFORMATION       269 

at  condenser  terminals 

E,  =  $  +  (a   +  j)xj 

=  {Z  +  (a+  j>i}/  (34) 


and  also 

E,  =  (b  -  j>0/i  (35) 

hence 

Z  +  (a  +  j)Xl 

/I=        (b-j>,      '  (36) 

and 

/O    =    J   +   /I 

=  Z  +  (b  -  j)x0  +  (a  +  j)xt 

(6  -  j>. 
_  Z  -  J(XQ  -  Xi)  +  (6x0  + 


(6  -  j>. 
and  the  impressed  e.m.f. 

e0  =  Ei  +  (a  +  j)z0/o; 
hence,  substituting  (35)  and  (37), 
_ 


,„_, 


b-j 

(38) 

Since  a  and   6  are  very  small  quantities,  their  products  and 
squares  can  be  neglected,  then 

XQ  +   [Z(a   +  b}  -  jxQ(a  -  b)  +  jgl(q  +  6)| 
^  _  •  ~ 

or 


zo  +  {Z(a  +  6)  -  jx0  (a  -  6)  +  j 
this  can  be  written 

1      +         6 


T      J^V      

X"    1  +   1  =;-  (a  +  6)  -  j(a  -  b)  +j?  (a  +  b) 


hence 


I  =  --     1  +ja-j(fl  +  b)  -     -  (a  +  b)  (41) 


that  is,  due  to  the  loss  of  power  in  the  reactances,  the  secondary 
current  is  less  than  it  would  be  otherwise,  and  decreases  with 
increasing  load  still  further. 


270  ELECTRIC  CIRCUITS 

Equation  (41)  can  also  be  written 


here  the  imaginary  component  is  very  small  in  the  parenthesis, 
that  is,  the  secondary  current  remains  practically  in  quadrature 
with  the  primary  voltage. 

The  absolute  value  is,  neglecting  terms  of  secondary  order, 


The  primary  current  is,  by  equation  (37)  and  (40), 
Z  —  J(XQ  —  Xi)  +  (bxQ 


XQ  +  Z(a  +  6)  -  jx0(a  -  b)  +  jx,(a  +  6)  XQ 


o_      ( 


€Q  XQ  XQ/  XQ 

XQ  i    i   %  ~  Jxi  /     i  TA       •  f        i\ 
1  +  -          -  (a  +  6)  -  3  (a  -  6) 

XQ 

139.  Example  — 

Considering  the  same  example  as  before:  a  constant-potential 
circuit  of  eQ  =  6600  volts  supplying  a  100-lamp  series  arc  circuit, 
with  i'  —  7.5  amp.,  and  ef  =  7500  volts  at  full-load  of  93  per  cent. 
power-factor,  that  is,  k  =  0.4,  and  Z  =  (1  —  0.4j)r.  Assuming 
now,  however,  the  loss  in  the  inductive  reactance  as  3  per  cent., 
and  in  the  capacity  as  1  per  cent.,  that  is,  a  =0.03  b  =0.01,  the 
full-load  value  of  the  secondary  load  impedance  is:  z'  =1000 
ohms,  rr  =930  ohms  and  x'  =372  ohms. 

To  give  non-inductive  primary  supply  at  full-load,  the  follow- 
ing equation  must  be  fulfilled  : 

Xi   =  XQ  —  X1   =  XQ  —  372. 

From  equation  (43),  the  secondary  current,  at  full-load,  is 

i>  =  *°  |  i  _  L  (a  + 

XQ(  XQ^ 


or 


.  .       6600  f ,        930  X  0.04 
7.5  =  {  1 


XQ     {  XQ 

hence 

XQ  =  840  ohms,  and  Xi  =  468  ohms. 


CONSTANT-CURRENT  TRANSFORMATION       271 


Substituting  in  (42),  (43),  (44), 


-  °'04 


84() 


8lo)  1 


,•-  7.86  (1-0.04  4) 
e  =  iz  =  1.077  ri 
=  8.46  r  (1-0.04^) 


0.4  r\ 
8407 


0.04 


/o  =  7.86 


and  herefrom  the  power-factor,  efficiency,  etc. 


FIG.  120. 

In  Fig.  120,  there  are  plotted,  with  the  secondary,  e.m.f.,  e, 
as  abscissae,  the  values:  secondary  current,  i;  primary  current,  IQ\ 
primary  power-factor,  cos  0,  and  efficiency. 

140.  In  alternating-current  circuits  small  variations  of  fre- 
quency are  unavoidable,  as  for  instance,  caused  by  changes  of 
load,  etc.,  and  the  inductive  reactance  is  directly  proportional,  the 
condensive  reactance  inversely  proportional  to  the  frequency. 
Wherever  inductive  and  condensive  reactances  are  used  in  series 
with  each  other  and  of  equal  or  approximately  equal  reactance, 
so  more  or  less  neutralizing  each  other,  even  small  changes  of 
frequency  may  cause  very  large  variations  in  the  result,  and  in 


272  ELECTRIC  CIRCUITS 

such  cases  it  is  therefore  necessary  to  investigate  the  effect  of  a 
change  of  frequency  on  the  result:  for  instance,  in  a  resonating 
circuit  of  very  small  power  loss,  a  small  change  of  frequency  at 
constant  impressed  e.m.f.  may  change  the  current  over  an  enor- 
mous range. 

Since  in  the  preceding,  constant-current  regulation  is  produced 
by  inductive  and  condensive  reactances  in  series  with  each  other, 
the  effect  of  a  variation  of  frequency  requires  investigation. 

Let,  then,  the  frequency  be  increased  by  a  small  fraction,  s. 

The  inductive  reactance  thereby  changes  to  0*0 (1  -f-  s)  and 
x(l  +  s),  and  Z  =  r  +  j(l  +  s)x  respectively,  and  the  conden- 

X0 

sive  reactance  to  , — ; — . 
l  +  « 

Leaving  all  the  other  denotations  the  same,  and  neglecting  the 
loss  of  power  in  the  reactances, 

E  =  ZI 


1  +V 
hence, 

/i  =3  —  '— -  / 

and 


XQ 

thus 


hence,  .expanding  and  dropping  terms  of  higher  order, 

eQ=+JI  [x0  +  s(x0  -  2a?i  +  4  Zj)-s2(3x! 
or 

7._J!?(l_.(l_2«!  +  4^)}.  (45) 

x0  \  \  X0  a;0  /  J 

Hence,  the  current  is  not  greatly  affected  by  a  change  of 
frequency.  That  is,  the  constant-current  regulation  of  the 
above-discussed  device  does  not  depend,  or  require,  a  constancy 
of  frequency  beyond  that  available  in  ordinary  alternating- 
current  circuits. 


CONSTANT-CURRENT  TRANSFORMATION       273 

B.  Monocyclic  Square 

141.  General. — 

A  combination  of  four  equal  reactances,  two  condensive  and 
two  inductive,  arranged  in  a  square  as  shown  diagrammatically 
in  Fig.  119,  page  261,  transforms  a  constant  voltage,  impressed 
upon  one  diagonal,  into  a  constant  current  across  the  other 
diagonal,  and  inversely. 

Let,  then, 

$o  =  eQ  =  constant  =  primary  impressed  e.m.f.,  or  supply 

voltage, 

$    =  secondary  terminal  voltage, 
$1  =  voltage  across  the  condensive 

reactance, 
$2  =  voltage  across  the  inductive 

reactance, 
and 

/o  :  --  primary  supply  current, 

/    =  secondary  current, 

/i  =  current  in  condensive  reactance, 

/2  =  current  in  inductive  reactance, 

these  currents  and  e.m.fs.  being  assumed  in  the  direction  as 
indicated  by  the  arrows  in  Fig.  119. 
Let 

x0  =  condensive  and  inductive  reactances; 
hence, 

Zi  =  —  jx0=  condensive  reactance  (1) 

Z2  =  +  jxQ  =  inductive  reactance  (2) 

Then,  at  the  dividing  points, 

/o  =  h  +  h  (3) 

and 

/    =/2-/i  (4) 

hence, 

/,  =  ^  W 

and 


/.  =  (6) 

In  the  e.m.f.  triangles, 

e^ZJi  +  ZJt  (7) 


18 


274  ELECTRIC  CIRCUITS 

and 

E  =  Zlll  -  Z2/2  (8) 

and 

E  =  ZI  (9) 

substituting  (1)  and  (2)  in  (7)  and  (8)  gives 

60=  -jzo(/i  ~/2)  (10) 

and 

ZI  =  -jz0(/i  +  /2)  (11) 

and,  substituting  herein  the  current, 

e0  =  +  jxol  (12) 

and 

ZI  =  -  jx0IQ  (13) 

hence,  the  secondary  current  is 

,._a      ,^,r  .     (14) 

the  primary  current, 

1|>--S       ;;;:;,\(15) 

the  condenser  current, 


and  the  current  in  the  inductive  reactance, 


The  secondary  terminal  voltage  is 

E  =  -  je0  £  (18) 

the  condenser  voltage, 

'"'  "^—  «o  (19) 


and  the  inductive  reactance  voltage, 

#.  =  +  jzo/i  =  +  J'(2~^  «„.  (20) 

-^  3?0 

The  tangent  of  the  primary  phase  angle  is 

tan  00  =  -  =  tan  0  (21) 

hence,  the  absolute  value  of  the  secondary  current  is 

(22) 


CONSTANT-CURRENT  TRANSFORMATION       275 

of  the  primary  current, 

*'° =  5  (23) 

of  the  condenser  current, 

•  *\/  y*     ~r"   v**^0  ~T~  **'/  /o  A  \ 

1 1  =  -  2      2          -  eo  (24) 

and  of  the  inductive  reactance  current 

.  \/  T     ~f~   \Xo          X)  /OK^ 

2  2    =    ^ o e°'  **°' 

&  XQ 

The  secondary  terminal  voltage  is 


e  =  -  eo  (26) 

X0 


the  condenser  voltage, 


Vr2  +  (g0  +  op2 


2z0 
and  the  inductive  reactance  voltage, 


(x0  -  xY  p  ,^ 

eQ.  (28) 


2  XQ 

142.  From  these  equations  follow  the  apparent  powers,  or  volt- 
amperes  of  the  different  circuits  as: 
Output, 

QQ  =  ei  =  ^?.  (29) 

Input, 

•  ^'  (30) 


Hence  the  input  is  the  same  as  the  output.     This  is  obvious, 
since  the  losses  of  power  in  the  reactances  are  neglected,  and  it 
was  found  (21),  that  the  phase  angle  or  the  power-factor  of  the 
primary  circuit  equals  that  of  the  secondary  circuit. 
Apparent  power  of  the  condensive  reactance, 

«,  -  arfl  =  ^±£±£*-V  (31) 

Inductance, 

n        0  ;        r*  +  fa  -~*)2     2. 
V2  =  e2*2  =  ~    ~  e°  > 


and,  therefore,  total  volt-ampere  capacity  of  the  reactances  is 

Q  =  2  (d  +  Q2) 


276  ELECTRIC  CIRCUITS 

=  r2  4  x2  4  so2     2. 
hence 

Q==Z*^/°2 e°2  (33) 

and, 
apparatus  economy, 

/-|-^f^i  (34) 

hence  a  maximum  for  2;  =  XQ  (35) 

and  this  maximum  is  equal  to  /0  =  M,or  50  per  cent.  (36) 

That  is,  the  maximum  apparatus  economy  of  the  monocyclic 
square,  as  discussed  here,  is  50  per  cent.,  or  in  other  words,  for 
every  kilovolt-ampere  output,  2  kv.-amp.  in  reactances  have  to 
be  provided. 

This  apparatus  economy  is  higher  than  that  of  the  T-connec- 
tion,  in  which  under  the  same  conditions,  that  is  for  Xi  =  x0,  the 
apparatus  economy  was  only  25  per  cent. 

The  commercial,  or  cost  economy  would  be  given  by 

a  = _^ =  maximum  (37) 

v  c\     /        y"v        •  s^  "^ 


where 

HI  =  price  per  kilovolt-ampere  of  condensive  reactance,  n2  = 
price  per  kilovolt-ampere  of  inductive  reactance. 
143.  Example. — 

Considering  the  same  problem  as  under  A.     From  a  constant 
impressed  e.m.f.  eQ  =  6600  volts,  a  100-lamp  arc  circuit,  of  93 
per  cent,  power-factor,  is  to  be  operated,  requiring 
i  =  7.5  amp. 
Z  =  r  +  jx 

=  r  (1  4  jk) 
where 

k  =  *  =  0.4; 

hence 

Z  =  r  (1  4  0.4  j), 
and  at  full-load 

e'  =  7500  volts. 
Then,  from  (22), 

Xo  =  f2  =  880  ohms,         z'  =  ^  =  1000  ohms; 


CONSTANT-CURRENT  TRANSFORMATION       277 

hence 

r'  =  930  ohms,  x'  =  kr'  =  372  ohms, 

and,  therefore, 

i  =  7.5  amp., 

{°  =  7"5  slo  amp'f 
e  =  7.5  z, 

and  at  full-load,  or  r  =  930,  when  denoting  full-load  values  by 
prime, 

i'      =  7.5  amp., 

i'o     =  7.93  amp., 

i'\     =  6.65  amp., 

i'z     =  4.52  amp., 

e'      =  7500  volts, 

e'i     =  5850  volts, 

e'2     =  3980  volts, 

£/*  =     j  56.25  kv.-amp. 

r  ao  - 

P'a  =  38.9  kv.-amp. 
P'a  =  18.0  kv.-amp. 
P'a  =  H3.8  kv.-amp. 

/'  =  0.4943 

or  49.43  per  cent,  that  is,  practically  the  maximum. 

144.  Power  Loss  in  Reactances. — 

In  the  preceding,  as  first  approximation,  the  loss  of  power  in 
the  reactances  has  been  neglected,  and  so  the  constancy  of 
current,  i,  was  perfect,  and  the  output  equal  to  the  input.  Con- 
sidering, however,  the  loss  of  power  in  the  reactances,  it  is 
found  that  the  current,  i,  varies  slightly,  decreasing  with  increas- 
ing load,  and  the  input  exceeds  the  output. 

Let,  then, 

Zi  =  (b  —  j)  x0  =  condensive  reactance, 
Z%  =  (a  -\-  j)  XQ  =  inductive  reactance, 

otherwise  retaining  the  same  denotations  as  in  the   preceding 
paragraphs, 

Then,  substituting  in  (7)  and  (8), 

^  =  (&-j)/i  +  («+J)/i  (38) 

XQ 

=  (&-.?)/i-  («+j)/i  (39) 


278  ELECTRIC  CIRCUITS 

Assuming 


Cj  =  ^p,  c2  =  =-ji  (41) 

Substituting  in  (38)  and  (39) 

60  .XT  T     X          ,  /T  ,          T     N  ,f  j     X 

^0 

substituting  herein  from  equations  (3)  and  (4)  gives 

-  =  (^2  +  j)/  +  CI/Q  (42) 

and 

~  =  -  c,/  -  (ca  +  j)/o  (43) 

and  from  these  two  equations  with  the  two  variables,  7  and  70, 
it  follows  from  (43)  that 

/^       "~~      vl«l/0       X  /  jt    A  \ 

o  =  -  TT^T rf  /  (44) 

w     i    C%)XQ 

Substituting  (44)  in  (42),  transposing,  and  dropping  terms  of 
secondary  order,  that  is,  products  and  squares  of  c\  and  Cz,  gives 


+  jc2-Cl^j  (45) 

substituting  (45)  in  (44),  and  transposing, 

then,  substituting  (45)  and  (46)  in  (5)  and  (6), 


and  the  absolute  value  is 


(48) 


--l°(l-  „,*•),  etc.  (49) 

XQ  \  T  I 


CONSTANT-CURRENT  TRANSFORMATION       279 

or,  approximately, 

,-- 

XQ 

145.  Example.  — 

Considering  the  same  example  as  before,  of  a  7.5-amp.  100- 
lamp  arc  circuit  operated  from  a  6600-volt  constant-potential 
supply,  and  assuming  again  as  in  paragraph  139: 

3  per  cent,  power-factor  of  inductive  reactance,  or  a  =  0.03. 

1  per  cent,  power-factor  of  condensive  reactance,  or  b  =  0.01. 
It  is  then, 

d  =  0.02,  c2  =  0.01, 
and  at  full-load, 


XQ  XQ 

or, 


hence, 

x0  =  861,  and  i  =  7.66  (l  -  0.02 
and  we  have,  approximately, 


7-66          +  0.02 


e  =  zi  =  1.077  n. 

In  Fig.  121  are  plotted,  with  the  secondary  terminal  voltage,  e, 
as  abscissae,  the  values  of  secondary  current,  i]  primary  current, 
to;  condenser  current,  iij  inductive  reactance  current,  i^  and 
efficiency. 

As  seen,  with  the  monocyclic  square,  the  current  regulation 
is  closer,  and  the  efficiency  higher  than  with  the  T  connection. 
This  is  due  to  the  lesser  amount  of  reactance  required  with  the 
monocyclic  square. 

The  investigation  of  the  effect  of  a  variation  of  frequency 
on  the  current  regulation  by  the  monocyclic  square,  now  can 
be  carried  out  in  the  analogous  manner  as  in  A  with  the  T 
connection. 


280  ELECTRIC  CIRCUITS 

C.     General  Discussion  of  Constant-potential  Constant- 
current  Transformation 

146.  In  the  preceding  methods  of  transformation  between 
constant  potential  and  constant  current  by  reactances,  that  is, 
by  combinations  of  inductive  and  condensive  reactances,  the 
constant  alternating  current  is  in  quadrature  with  the  constant 
e.m.f.  Even  in  constant-current  control  by  series  inductive 
reactances,  the  constancy  of  current  is  most  perfect  for  light 
loads,  where  the  reactance  voltage  is  large  and  thus  the  constant- 
current  voltage  almost  in  quadrature,  and  the  constant-current 
control  is  impaired  in  direct  proportion  to  the  shift  of  phase 
of  the  constant  current  from  quadrature  relation. 


FIG.  121. 

The  cause  hereof  is  the  storage  of  energy  required  to  change 
the  character  of  the  flow  of  energy.  That  is,  the  energy  supplied 
at  constant  potential  in  the  primary  circuit,  is  stored  in  the  react- 
ances, and  returned  at  constant  current,  in  the  secondary  circuit. 

The  storage  of  the  total  transformed  energy  in  the  reactances 
allows  a  determination  of  the  theoretical  minimum  of  reactive 
power,  that  is,  of  inductive  and  condensive  reactances  required 
for  constant-potential  to  constant-current  transformation,  since 
the  energy  supplied  in  the  constant-current  circuit  must  be  stored 
for  a  quarter  period  after  being  received  from  the  constant-po- 
tential circuit. 


CONSTANT-CURRENT  TRANSFORMATION       281 

Let 

p   =  P(l  +  cos  2  0) 

=    Power   supplied   to   the   constant-current   circuit; 

thus,  neglecting  losses, 
pQ  =  P(l  -  cos  2  0) 

=  Power  consumed  from  the  constant-potential  cir- 
cuit, and 
p0  -  p  =  2  P  cos  2  e 

=  Power  in  the  reactances. 

That  is,  to  produce  the  constant-current  power,  P,  from  a 
single-phase  constant-potential  circuit,  the  apparent  power,  2  P, 
must  be  used  in  reactances;  or,  in  other  words,  per  kilowatt  con- 
stant-current power  produced  from  a  single-phase  constant-po- 
tential circuit,  reactances  rated  at  2  kv.-amp.  as  a  minimum  are 
required,  arranged  so  as  to  be  shifted  45°  against  the  constant- 
potential  and  the  constant-current  circuit. 

The  reactances  used  for  the  constant-potential  constant-cur- 
rent transformation  may  be  divided  between  inductive  and  con- 
densive  reactances  in  any  desired  proportion. 

The  additional  wattless  component  of  constant-potential 
power  is  obviously  the  difference  between  the  wattless  volt-am- 
peres of  the  inductive  and  that  of  the  condensive  reactances. 
That  is,  if  the  wattless  volt-amperes  of  reactance  is  one-half  of 
inductive  and  one-half  of  condensive,  the  resultant  wattless  volt- 
amperes  of  the  main  circuit  is  zero,  and  the  constant-potential 
circuit  is  non-inductive,  at  non-inductive  load,  or  consumes  cur- 
rent proportional  to  the  load. 

If  A  is  the  condensive  and  B  the  inductive  volt-amperes,  the 
resultant  wattless  volt-amperes  is  B-A;  that  is,  a  lagging  watt- 
less volt-amperes  of  B-A  (or  a  leading  volt-ampere  of  A-B) 
exist  in  the  main  circuit,  in  addition  to  the  wattless  volt-amperes 
of  the  secondary  circuit,  which  reappear  in  the  primary  circuit. 

147.  These  theoretical  considerations  permit  the  criticism  of 
the  different  methods  of  constant-potential  to  constant-current 
transformation  in  regard  to  what  may  be  called  their  apparatus 
economy,  that  is,  the  kilovolt-ampere  rating  of  the  reactance  used, 
compared  with  the  theoretical  minimum  rating  required. 

1.  Series  inductive  reactance,  that  is,  a  reactive  coil  of  constant 
inductive  reactance  in  series  with  the  circuit.  This  arrangement 
obviously  gives  only  imperfect  constant-current  control.  Per- 


282  ELECTRIC  CIRCUITS 

mitting  a  variation  of  5  per  cent,  in  the  value  of  the  current  (that 
is,  full-load  current  in  5  per  cent,  less  than  no-load  current)  and 
assuming  4  per  cent,  loss  in  the  reactive  coil,  a  reactance  rated  at 
2.45  kv.-amp.  is  required  per  kilowatt  constant-current  load. 
This  apparatus  operates  at  87.9  per  cent,  economy  and  30  per 
cent,  power-factor. 

Assuming  10  per  cent,  variation  in  the  value  of  the  current, 
reactance  rated  at  2.22  kv.-amp.  is  required  per  kilowatt  constant- 
current  load.  This  arrangement  operates  at  an  economy  of  91.8 
per  cent.,  and  a  power-factor  of  49.5  per  cent. 

In  the  first  case,  the  apparatus  economy,  that  is,  the  ratio  of 
the  theoretical  minimum  kilovolt-ampere  rating  of  the  reactance 
to  the  actual  rating  of  the  reactance  is  88  per  cent.,  and  in  the  last 
case  92  per  cent.,  thus  the  objection  to  this  method  is  not  the  high 
rating  of  the  reactance  and  the  economy,  but  the  poor  constant- 
current  control,  and  especially  the  very  low  power-factor. 

2.  Inductive  and  condensive  reactances  in  resonance  condition, 
the  condensive  reactance  being  shunted  by  the  constant-current 
circuit.     In  this  case,  condensive  reactance  rated  at  1  kv.-amp. 
and  inductive  reactance  rated  at  2  kv.-amp.  are  required  per  kilo- 
watt constant-current  load,  and  the  main  circuit  gives  a  constant 
wattless  lagging  apparent  power  of  1  kv.-amp.     Assuming  again 
4  per  cent,  loss  in  the  inductive  and  2  per  cent,  loss  in  the  condens- 
ive reactances,  gives  a  full-load  efficiency  of  91  per  cent,  and  a 
power-factor  (lagging)  of  74  per  cent.     The  apparatus  economy 
by  this  method  is  66.7  per  cent. 

3.  Inductive  and  condensive  reactances  in  resonance  condition, 
the  inductive  reactance  shunted  by  the  constant-current  circuit. 
In  this  case,  as  a  minimum,  per  kilowatt  constant-current  load, 
condensive  reactance  rated  at  2  kv.-amp.  and  inductive  reactance 
rated  at  1  kv.-amp.  is  required,  and  the  main  circuit  gives  a  con- 
stant wattless  leading  apparent  power  of  1  kv.-amp.     The  effi- 
ciency of  transformation  is  at  full-load  92.5  per  cent.,  the  power- 
factor  (leading)  73  per  cent.,  the  apparatus  economy  66.7  per  cent. 

4.  T-connection,  that  is,  two  equal  inductive  reactances  in  se- 
ries to  the  constant-current  circuit  and  shunted  midway  by  an 
equal  condensive  reactance.     In  this  case  per  kilowatt  constant- 
current  load,  condensive  reactance  rated  at  2  kv.-amp.  and  in- 
ductive reactance  rated  at  2  kv.-amp.  are  required. 

The  main  circuit  is  non-inductive  at  all  non-inductive  loads, 
that  is,  the  power-factor  is  100  per  cent. 


CONSTANT-CURRENT  TRANSFORMATION       283 

The  full-load  efficiency  is  89.3  per  cent,  apparatus  economy  50 
per  cent. 

5.  The  monocyclic  square.     In  this  case  a  condensive  reactance 
rated  at  1  kv.-amp.  and  inductive  reactance  rated  at  1  kv.-amp. 
are  required  per  kilowatt  constant-current  load.     The  main  cir- 
cuit is  non-inductive  at  all  non-inductive  loads,  that  is,  the  power- 
factor  is  100  per  cent.     The  fulWoad  efficiency  is  94.3  per  cent., 
the  apparatus  economy  100  per  cent. 

6.  The   monocyclic  square  in   combination   with   a   constant- 
potential  polyphase  system  of  impressed  e.m.f.     In  this  case,  per 
kilowatt  constant-current  load,  condensive  reactance  rated  at  0.5 
kv.-amp.  and  inductive  reactance  rated  at  0.5  kv.-amp.  are  re- 
quired.    The  main  circuits  are  non-inductive  at  all  loads,  that  is, 
the  power-factor  is  100  per  cent.     The  full-load  efficiency  is  over 
97  per  cent,  the  apparatus  economy  200  per  cent. 

148.  In  the  preceding,  the  constant-potential  to  constant-cur- 
rent transformation  with  a  single-phase  system  of  constant  im- 
pressed e.m.f.,  has  been  discussed;  and  shown  that  as  a  minimum 
in  this  case,  to  produce  1  kw.  constant-current  output,  reactances 
rated  at  2  kv.-amp.  are  required  for  energy  storage.  The  con- 
stant current  is  in  quadrature  with  the  main  or  impressed  e.m.f., 
but  can  be  either  leading  or  lagging.  Thus  the  total  range  avail- 
able is  from  1  kw.  leading,  to  zero,  to  1  kw.  lagging.  Hence  if  a 
constant-quadrature  e.m.f.  is  available  by  the  use  of  a  poly- 
phase system,  the  range  of  constant  current  can  be  doubled,  that 
is,  reactance  rated  at  2  kv.-amp.  can  be  made  to  control  the  po- 
tential for  2  kw.  constant-current  output  in  the  way  shown  in  Fig. 
122  for  a  three-phase,  and  Fig.  123  for  a  quarter-phase  system  of 
impressed  e.m.f. 

In  this  case,  one  transformer  feeds  a  monocyclic  square,  the 
other  transformer  inserts  an  equal  constant  e.m.f.  in  quadrature 
with  the  former,  which  from  no-load  to  half-load  is  subtractive, 
from  half-load  to  full-load  is  additive,  that  is,  at  full-load,  both 
phases  are  equally  loaded;  at  half -load  only  one  phase  is  loaded 
and  at  no-load  one  phase  transforms  energy  into  the  other  phase. 

The  monocyclic  e.m.f.  square  in  this  case,  when  passing 
from  full-load  to  no-load,  gradually  collapses  to  a  straight  line 
at  half-load,  then  overturns  and  opens  again  to  a  square  in  the 
opposite  direction  at  no-load.  That  is,  at  full-load  the  trans- 
formation is  from  constant  potential  to  constant  current,  and  at 


284 


ELECTRIC  CIRCUITS 


no-load  the  transformation  is  from  constant  current  to  constant 
potential. 

Obviously  with  this  arrangement  the  efficiency  is  greatly 
increased  by  the  reduction  of  the  losses  to  one-half,  and  the  con- 
stant-current control  improved. 


Fia  122. 

At  the  same  time,  the  sensitiveness  of  the  arrangement  for  dis- 
tortion of  the  wave  shape,  as  will  be  discussed  later,  is  greatly 
reduced,  due  to  the  insertion  of  a  constant-potential  e.m.f.  into 
the  constant-current  circuit. 

Obviously  the  arrangments  in  Figs.  122  and  123  are  not  the 
only  ones,  but  many  arrangements  of  inserting  a  constant-quad- 
rature e.m.f.  into  the  monocyclic  square  or  triangle  are  suitable. 


FIG.  123. 

Different  arrangements  can  also  be  used  of  the  constant-current 
control,  for  instance,  the  inductive  and  condensive  reactances  in 
resonance  condition  with  their  common  connection  connected 
to  the  center  of  an  autotransformer  or  transformer,  with  the 
insertion  of  the  constant-potential  quadrature  e.m.f.  in  the  latter 
circuit  as  shown  in  Fig.  124,  or  the  T-connection,  shown  applied 
to  a  quarter-phase  system  in  Fig.  125. 


CONSTANT-CURRENT  TRANSFORMATION       285 


Constant-potential  apparatus  and  constant-current  single- 
phase  circuits  can  also  be  operated  from  the  same  transformer 
secondaries  in  a  similar  manner,  as  indicated  in  Fig.  124  for  a 
three-phase  secondary  system. 

In  Figs.  122  to  125  the  arrangement  has  been  shown  as  applied 
to  step-down  transformers,  but  in  the  estimate  of  the  efficiency 


ill 


D 

^x 

»-   i-  °- 

§™     K     S 
o  2  J 

H 

_J 

^> 

3> 
D 
^> 

CON8TANT  CURRENT      ' 
SINGLE-PHASE 

FIG.  124. 

the  losses  in  these  transformers  have  not  been  included,  since 
these  transformers  are  obviously  not  essential  but  merely  for  the 
convenience  of  separating  electrically  the  constant-current  cir- 
cuit from  the  high-potential  line.  It  is  evident,  for  instance,  in 
Fig.  124,  that  the  constant-current  and  constant-potential  cir- 


FIG.  125. 


cuits  instead  of  being  operated  from  the  three-phase  secondaries 
of  the  step-down  transformers  can  be  operated  directly  from  the 
three-phase  primaries  by  replacing  the  central  connection  of 
the  one  transformer  by  the  central  connection  of  the  auto- 
transformer. 


286  ELECTRIC  CIRCUITS 

D.  Problems 

149.  In  the  following  problems  referring  to  constant-potential 
to  constant-current  transformation  by  reactances,  it  is  recom- 
mended: 

(a)  To  derive  the  equation  of  all  the  currents  and  e.m.fs.,  in 
complex  quantities  as  well  as  in  absolute  terms,  while  neglecting 
the  loss  of  power  in  the  reactances. 

(6)  To  determine  the  volt-amperes  in  the  different  parts  of 
the  circuit,  as  load,  reactances,  etc.,  and  therefrom  derive  the 
apparatus  economy,  to  find  its  maximum  value,  and  on  which 
condition  it  depends. 

(c)  To  determine  the  effect  of  inductive  load  on  the  power 
of  the  primary  supply  circuit,  to  investigate  the  phase  angle 
of  the  primary  supply  circuit,  and  the  conditions  under  which 
it  becomes  a  minimum,  or  the  primary  supply  becomes  non- 
inductive. 

(d)  To  redetermine  the  equations  of  the  problem,  while  con- 
sidering the  power  lost  in  the  reactances,  and  apply  these  equa- 
tions to  a  numerical  example,  plotting  all  the  interesting  values. 

(e)  To  investigate  the  effect  of  a  change  of  frequency  on  the 
equations,  more  particularly  on  the  constant-current  regulation. 

(/)  To  investigate  the  effect  of  distortion  of  wave  shape, 
that  is,  the  existence  of  higher  harmonics  in  the  impressed 
e.m.f.,  and  their  suppression  or  reappearance  in  the  secondary 
circuit. 

(g)  To  study  the  reversibility  of  the  problem,  that  is,  apply 
(a)  to  (/)  to  the  reversed  problem  of  transformation  from  constant 
current  to  constant  potential. 

Some  of  the  transforming  devices  between  constant  potential 
and  constant  current  are: 

A.  Single-phase. 

(a)  The  resonating  circuit,  or  condensive  and  inductive 
reactances,  of  equal  values,  in  series  with  each  other  in  the  con- 
stant-potential circuit,  and  the  one  reactance  shunted  by  the 
constant-current  circuit. 

(6)  T-connection,  as  partially  discussed  in  (A). 

(c)  The  monocyclic  square,  as  partially  discussed  in  (B) . 

(d)  The  monocyclic  triangle:  a  condensive  reactance  and  an 
inductive  reactance  of  equal  values,  in  series  with  each  other 
across    the    constant-potential    circuit,    the    constant-current 


CONSTANT-CURRENT  TRANSFORMATION       287 

circuit  connecting  between  the  reactance  neutral,  or  the  common 
connection  between  the  two  (opposite)  reactances,  and  the 
neutral  of  a  compensator  or  autotransformer  connected  across 
the  constant-potential  circuit.  Instead  of  the  compensator 
neutral,  the  constant-current  circuit  can  be  carried  back  to  the 
neutral  of  the  transformer  connected  to  the  constant-potential 
circuit. 

B.  Polyphase. 

(a)  In  the  two-phase  system  the  two  phases  of  e.m.fs.,  e0 
and  je0,  are  connected  in  series  with  each  other,  giving  the 
outside  terminals,  A  and  B,  and  the  neutral  or  common  con- 
nection, C.  A  condensive  reactance  and  an  inductive  reactance 
of  equal  values,  in  series  with  each  other  and  with  their  neutral 
or  common  connection,  D,  are  connected  either  between  A  and 
B,  and  the  constant-current  circuit  between  C  and  D,  or  the 
reactances  are  connected  between  A  and  C,  and  the  constant- 
current  circuit  between  B  and  D.  In  either  case,  several  ar- 
rangements are  possible,  of  which  only  a  few  have  a  good  appara- 
tus economy. 

(6)  In  a  three-phase  system,  a  condensive  reactance,  an  induct- 
ive reactance  equal  in  value  to  that  of  the  condensive  reactance 
and  the  constant-current  circuit,  are  connected  in  star  connec- 
tion between  the  three-phase,  constant-potential  terminals. 
Here  also  two  arrangements  are  possible,  of  which  one  only 
gives  good  apparatus  economy. 

(c)  In  a  constant-potential  three-phase  system,  each  of  the 
three  terminals,  A,  B,  C,  connects  with  a  condensive  and  an 
inductive  reactance,  and  all  these  reactances  are  of  equal  value, 
and  joined  together  in  pairs  to  three  terminals,  a,  b,  c,  so  that 
each  of  these  terminals,  a,  6,  c,  connects  an  inductive  with  a 
condensive  reactance,  a,  6,  c,  then,  are  constant-current  three- 
phase  terminals,  that  is,  the  three  currents  at  a,  6,  c,  are  constant 
and  independent  of  the  load  or  the  distribution  of  load,  and 
displaced  from  each  by  one-third  of  a  period.  This  arrange- 
ment is  especially  suitable  for  rectification  of  the  constant  al- 
ternating-current, to  produce  constant  direct  current. 

150.  Some  further  problems  are : 

1.  In  a  single-phase,  constant-current  transforming  device, 
as  the  monocyclic  square,  the  constant  current,  i,  is  in  quadrature 
with  the  constant  impressed  e.m.f.,  e0.  By  inserting  a  constant- 
potential  e.m.f.,  $3,  into  the  constant-current  circuit,  the  appa- 


288  ELECTRIC  CIRCUITS 

ratus  economy  can  be  greatly  increased,  in  the  maximum  can 
be  doubled;  that  is,  the  e.m.f.,  E3  gives  constant-power  output, 
and  from  no-load  to  half -load,  the  transformation  is  from  con- 
stant current  to  constant  potential,  that  is,  a  part  of  the  power 
supply,  E3)  is  transformed  into  the  circuit,  of  e.m.f.,  e0,  that 
is,  the  circuit,  e0,  receives  power.  Above  half-load  the  circuit 
of  eo  transforms  power  from  constant  potential  to  constant 
current,  into  the  circuit  of  e.m.f.  Es. 

Since  i  is  in  time  quadrature  with  eo,  with  non-inductive 
secondary  load,  that  is,  the  secondary  terminal  voltage,  E,  in 
phase  with  the  secondary  current,  i,  E3  should  also  be  in  phase 
with  i,  that  is,  E3  =  jes.  With  inductive  secondary  load,  of 
phase  angle,  6,  E3  should  be  in  phase  with  Ej  that  is,  leading  i 
by  angle  6,  or  should  be:  E3  =  jes  (1  +  kf). 

It  is  interesting,  therefore,  to  investigate  how  the  equation  of 
the  constant-potential  constant-current  devices  are  changed 
by  the  introduction  of  such  an  e.m.f.,  E3,  at  non-inductive  as 
well  as  at  inductive  load,  if  E3  =  je3,  or  E3  —  j(e3  —  je'3),  in 
either  case,  and  also  to  determine  how  such  an  e.m.f.,  E3,  of 
the  proper  phase  relation,  can  be  derived  directly  or  by  trans- 
formation from  a  two-phase  or  three-phase  system. 

2.  If  in  the  constant-potential  constant-current  transform- 
ing device  one  of  the  reactances  is  gradually  changed,  increased 
or  decreased  from  its  proper  value,  then  in  either  case  the  regula- 
tion of  the  system  is  impaired.  That  is,  the  ratio  of  full-load 
current  to  no-load  current  falls  off,  but  at  the  same  time,  the 
no-load  current  also  changes. 

With  increase  of  load,  the  frequency  of  the  system  decreases, 
due  to  the  decreasing  speed  of  the  prime  mover,  if  the  output 
of  the  system  is  an  appreciable  part  of  the  rated  output.  If, 
therefore,  the  reactances  are  adjusted  for  equality  of  the  frequency 
of  full-load,  at  the  higher  frequency  of  no-load,  the  inductive 
reactance  is  increased,  and  thereby  the  no-load  current  decreased 
below  the  value  which  it  would  have  at  constant  reactance,  and 
in  this  manner  the  increase  of  current  from  full-load  to  no-load 
is  reduced. 

Such  a  drop  of  speed  and  therefore  of  frequency,  s,  can  there- 
fore be  found,  that  the  current  at  full-load,  with  perfect  equality 
between  the  reactances,  equals  the  current  at  no-load,  where 
the  reactances  are  not  quite  equal.  That  is,  the  variation  of 
frequency  compensates  for  the  incomplete  regulation  of  the 


CONSTANT-CURRENT  TRANSFORMATION       289 

current,  caused  by  the  energy  loss  in  the  reactances.  Further- 
more, with  a  given  variation  of  frequency,  s,  from  no-load  to  full- 
load,  the  reactances  can  be  chosen  so  as  to  be  slightly  unequal 
at  full-load,  and  more  unequal  at  no-load;  the  change  of  current 
caused  hereby  compensates  for  the  incomplete  current  regu- 
lation, that  is,  with  a  given  frequency  variation,  s  (within 
certain  limits),  the  current  regulation  can  be  made  perfect  from 
no-load  to  full-load,  by  the  proper  degree  of  inequality  of  the 
reactances. 

It  is  interesting  to  investigate  this,  and  apply  to  an  example, 
a,  to  determine  the  proper  s,  for  perfect  equality  of  reactance  at 
full-load;  6,  with  a  given  value  of  s  =  0.04,  to  determine  the  in- 
equality of  reactance  required.  Assuming  a  =  0.03;  b  =  0.01. 

3.  If  one  point  of  the  constant-current  circuit,  either  a 
terminal  or  an  intermediate  point,  connects  to  a  point  of  the 
constant-potential  circuit,  either  a  terminal  or  some  intermediate 
point  (as  inside  of  a  transformer  winding),  the  constant  current 
is  not  changed  hereby,  that  is,  the  regulation  of  the  system  is 
not  impaired,  and  no  current  exists  in  the  cross  between  the  two 
circuits.  The  distribution  of  potential  between  the  reactances, 
however,  may  be  considerably  changed,  some  reactances  re- 
ceiving a  higher,  others  a  lower  voltage. 

It  follows  herefrom,  that  a  ground  on  a  constant-current 
system  does  not  act  as  a  ground  on  the  constant-potential  system, 
but  electrically  the  two  systems,  although  connected  with  each 
other,  are  essentially  independent,  just  as  if  separated  from  each 
other  by  a  transformer.  So,  for  instance,  in  the  monocyclic 
square,  one  side  may  be  short-circuited  without  change  of 
current  in  the  secondary,  but  with  an  increase  of  current  in  the 
other  three  sides.  It  is  interesting  to  investigate  how  far  this 
independence  of  the  circuits  extends. 

In  general,  as  an  example,  the  following  constants  may  be 
chosen:  In  the  constant-potential  circuit:  e0  =  6600  volts  and 
i'o  =  10  amp.  at  full-load. 

In  the  constant-current  circuit:  i  =  7.5  amp.,  e'  =  7500  volts 
at  full-load. 

Or,  especially  in  polyphase  systems,  ef,  respectively,  i'0 
corresponding  to  the  maximum  economy  point, 

and  a  =  0.03;  b  =  0.01. 


19 


290  ELECTRIC  CIRCUITS 

E.  Distortion  of  Voltage  Wave 

151.  It  is  of  interest  to  investigate  what  effect  the  distortion 
of  the  voltage  wave,  that  is,  the  existence  of  higher  harmonics 
in  the  wave  of  supply  voltage,  has  on  the  regulation  of  the  con- 
stant-potential   constant-current    transformation    systems    dis- 
cussed in  the  preceding. 

Where  constant  current  is  produced  by  inductive  reactance 
only,  higher  harmonics  in  the  voltage  wave  naturally  are  sup- 
pressed the  more,  the  larger  the  inductive  reactance  and  the  higher 
the  order  of  the  harmonic. 

An  increase  of  the  intensity  of  the  harmonics  in  the  current 
wave,  over  that  in  the  voltage  wave,  and  with  it  an  impairment 
of  the  constant-current  regulation,  could  thus  be  expected  only 
with  devices  using  capacity  reactance. 

As  example  may  be  investigated  the  effect  of  the  distortion 
of  the  impressed  voltage  wave  on  the  T  connection,  and  on  the 
monocyclic  square. 

The  symbolic  method  of  treating  general  alternating  waves 
may  be  used,  as  discussed  in  Chapter  XXVII,  of  "Theory  and 
Calculation  of  Alternating-current  Phenomena,"  fifth  edition, 

page  379.     That  is,  the  voltage  wave  is  represented  by 

oo 

1 
and  the  impedance  by 

Z  =  r  +  jn  (nxm  +  x0  H-  -H 

IV   / 

where 

n  =  order  of  harmonic. 

A.     T  Connection  or  Resonating  Circuit 

152.  Assuming  the  same  denotation  as  before,  we  have,  for 
the  nth  harmonic: 

primary  inductive  reactance, 

ZQ   =    -f-  JUXQ', 

secondary  inductive  reactance, 

Zi  =  +jnxi; 

condensive  reactance, 

jzo. 


CONSTANT-CURRENT  TRANSFORMATION       291 

when  neglecting  the  energy  losses  in  the  reactances, 
load 

Z  =  r(l+jnk) 

therefore,  also  for  the  nth  harmonic. 


T7T  77T 

JjJ  1     ==     XV 

and  also 


hence, 

=  .  n[r  (1  +  jnfc)  +  jnxi] 

and 

/«    =    /+/, 


r. 

hence, 


=  I  [r  (1  +  jnfc)  +  jnajj  +  n  [jx0  -jnzxi  -  nr  (1  +  jnk)\  u 
=  {  -  (n2  -l)r(l  +  jnk)  -  jnx,  (n2  -  1)  +  jnx0}/; 
hence, 


nxo  —  nx\(n2  —  1)  +  j  (n2  —  1)  r  (1 

=  _  ~  J^o 

nxo  -  (n2  -  l)[n  (xl  +  kr)  +  jr]' 

hence,  approximately,  for  higher  values  of  n, 


that  is,  for  larger  values  of  n,  /  =  0,  or  the  higher  harmonics  in 
the  current  wave  disappear. 

Herefrom,  by  substituting  in  the  preceding  equations,  the 
supply  current,  /o,  the  condenser  current,  /i,  their  respective 
e.m.fs.,  etc.,  are  derived. 

It  is  then,  in  general  expression : 

If 


(en  —  jnCn1)  =  impressed  e.m.f., 

i7 


292  ELECTRIC  CIRCUITS 

j*  i   (P     _i_  i   P   i\ 

^•^  Jn\^n      I     Jn^n   ) 

I  '-  =  &  nx0  -  (n2  -  1)  [nfa  +  kr)  +  jr] 

(en  -  jenl) 


the  equation  of  the  secondary  current. 
For  instance,  let 

#o   =   6600   ih   --   0.203   --   0.155  +  0.067   -   0.25 

=  constant-impressed  e.m.f. 
or,  absolute, 

eQ  =  6600  \1  +  0.202  +  0.152  +  0.062  +  0.252 
=  6600  X  1.062 
=  7010  volts, 

and  choosing  the  same  values  as  before,  in  paragraph  143, 

XQ  =  880  ohms, 
Xi  =  508  ohms, 
r'  =  930  ohms, 
k  =  0.4; 
it  is,  substituting, 

,_.  .   60.0.3  -  48.8  J3-  8.0  J5  +  1.2J7 


7  "  --7-5  508  +  0.4  r 


or,  absolute, 


,    2       60.02  +  48.82  +  8.02  +  1.22 
(508  +  0.4  r)2 


604,600 


1    (508 -j- 0.4  r)2' 
hence,  at  no-load, 

i  =  7.5  X  1.00021 
and,  at  full-load, 

r  =  930, 

i  =  7.5  X  1.00003. 

That  is,  the  current  wave  is  as  perfect  a  sine  wave  as  possible, 
regardless  of  the  distortion  of  the  impressed  e.m.f.,  which,  for 
instance,  in  the  above  example,  contains  a  third  harmonic  of  32 
per  cent.  Or  in  other  words,  in  the  T  connection  or  the  resonat- 
ing circuit,  all  harmonics  of  e.m.f.  are  wiped  out  in  the  current 
wave,  and  this  method  indeed  offers  the  best  and  most  conven- 
ient means  of  producing  perfect  sine  waves  of  current  from  any 
shape  of  e.m.f.  waves. 


CONSTANT-CURRENT  TRANSFORMATION        293 

153.  B.  Monocyclic   Square 

Assuming  the  same  denotation  as  before,  we  have  for  the  nth 
harmonic  : 

inductive  reactance, 

Zz  =  +  jnxQ; 
condensive  reactance, 


load, 
currents, 


and 


e.m.fs., 

#0    = 

Z7  = 

hence,  substituting,  we  have 


=  -  ^0(^  +  n/2) 
thus, 


then,  combining,  we  obtain 


and 


2  *o  +  jr(l  +  jnfr)  (n  -  ^ 

-  JE0  (n2  +  1) 


jnk)' 


and  herefrom  /i,  /i,  /2,  etc. 


294  ELECTRIC  CIRC V ITS 

Approximately,  for  higher  values  of  n,  and  for  high  loads,  rt 

7  ^L° 

nkr 

That  is,  the  higher  harmonics  of  current  decrease  proportion- 
ally to  their  order,  at  heavy  loads — that  is,  large  values  of  r.  For 
light  loads,  however,  or  small  values  of  r,  and  in  the  extreme  case, 
at  no-load,  or  r  =  0,  it  is 

=        JE»  (n2  +  1) 

2nx<> 
and,  approximately, 

_  jE0n 
~  2xQ' 

That  is,  the  current  is  increased  proportional  to  the  order  of  the 
harmonics,  or  in  other  words,  at  no-load,  in  the  monocyclic  square, 
the  higher  harmonics  of  impressed  e.m.fs.  produce  increased 
values  of  the  higher  harmonics  of  current,  that  is,  the  wave-shape 
distortion  is  increased  the  more,  the  higher  the  harmonics. 

In  general  expression : 

If 

00 

$0  =  X^n  ~~  -?ne'n)  =  impressed  e.m.f., 

i 

T  =  v  _    Jn(n*  +  i)(en-jne'n) 

"  ^2  nx0  +  jnr(n2  -!)(!+  jnnk) 

and  herefrom  /0,  /i,  /2,  etc. 
For  instance,  let 

Eo  =  6600   {li  -  0.203  -  0.25  j3  -  0.155  +  0.067) 

=  constant-impressed  e.m.f., 
or,  absolute, 

e0  =  7010  volts, 

and,  choosing  the  same  values  as  before, 
xQ  =   880  ohms, 
/   =   930  ohms, 

k  =  0.4; 
it  is,  substituted, 

(2.5  -  2j3)6600  25,740 

5280  -  (9.6  -  8  j)r       8800  -  (48  -  24  j6)r 

19,800 . 

h  12,320  -  (134.4  -48j)r; 


CONSTANT-CURRENT  TRANSFORMATION       295 

herefrom  follows, 

at  no-load,  r  =  0, 

/   =  7.5  -  (3.12  -  2.5  jj)  -  2.92  +  1.617. 

That  is,  at  no-load,  the  secondary  current  contains  excessive 
higher  harmonics,  for  instance,  a  third  harmonic, 


V  3.122  +  2.52  =  4.0,  or  53.3  per  cent,  of  the  fundamental. 
Absolute,  the  no-load  current  is 


*  ==  V  7.52  +  3.122  +  2.52  +  2.922  +  1.612  =  9.13    amp. 
At  full-load,  or  r  =  930,  it  is 
/  =  7.5  +  (2.18  +  1.07  js)  +  (0.51  +  0.32  J5)  -  (0.14  +  0.06  j7); 

that  is,  at  full-load,  the  harmonics,  while  still  intensified,  are  less 
than  at  no-load,  and  decrease  with  their  order,  n,  more  rapidly. 
The  absolute  value  is 

i  =  V7.52  +  2.182  +  1.07*  +  0.512  +  0.322+  0.142  +  0.062 
=  7.91  amp. 

Instead  of  7.5  amp.,  the  value  which  the  current  would  have 
at  all  loads  if  no  higher  harmonics  were  present,  the  higher  har- 
monics of  impressed  e.m.f. -raise  the  current  to  9.13  amp.,  or  by 
21.7  per  cent,  at  no-load,  and  to  7.91  amp.,  or  by  5.5  per  cent,  at 
full-load,  while  the  impressed  e.m.f.  is  increased  by  6.2  per  cent, 
by  its  higher  harmonics. 

It  follows  also  that  the  constant-current  regulation  of  the  sys- 
tem is  seriously  impaired,  and  between  no-load  and  full-load  the 
current  decreases  from  9.13  to  7.91  amp.,  or  by  15.4  per  cent., 
which  as  a  rule  is  too  much  for  an  arc  circuit. 

154.  It  follows  herefrom : 

While  the  T  connection  of  transformation  from  constant  poten- 
tial to  constant  current  suppresses  the  higher  harmonics  of  im- 
pressed e.m.f.  and  makes  the  constant  current  a  perfect  sine  wave, 
the  monocyclic  square  intensifies  the  higher  harmonics  so  that  the 
higher  harmonics  of  impressed  e.m.f.  appear  at  greatly  increased 
intensity  in  the  constant-current  wave.  The  increase  of  the 
higher  harmonics  is  different  for  the  different  harmonics  and  for 
different  loads,  and  the  distortion  of  wave  shape  produced  hereby 
is  far  greater  at  no-load,  and  the  constant-current  regulation 
of  the  system  is  thereby  greatly  impaired,  and  at  load  the  dis- 


296  ELECTRIC  CIRCUITS 

tortion  is  less,  and  very  high  harmonics  are  fairly  well  sup- 
pressed, and  the  operation  of  an  arc  circuit  so  feasible. 

Assuming,  then,  that  in  the  monocyclic  square  of  constant- 
potential  constant-current  transformation,  with  a  distorted  wave 
of  impressed  e.m.f.,  we  insert  in  series  to  the  monocyclic  square 
into  the  main  circuit,  70,  two  reactances  of  opposite  sign,  which 
are  equal  to  each  other  for  the  fundamental  frequency, 

that  is,  a  condensive  reactance,  Z3  =   —  j—,  and  an  inductive 

/& 

reactance,  Z4  =  +  jnxs.  Then  for  the  fundamental,  these  two 
reactances  together  offer  no  resultant  impedance,  but  neutralize 
each  other,  and  the  only  drop  of  voltage  produced  by  them  is  that 
due  to  the  small  loss  of  power  in  them.  At  the  nth  harmonic, 
however,  the  resultant  reactance  is 


or,  approximately, 

and  two  such  impedances  so  obstruct  the  higher  harmonics,  the 
more,  the  higher  their  order  while  passing  the  fundamental  sine 
wave. 

Such  a  pair  of  equal  reactances  of  opposite  sign  so  can  be  called 
a  "wave  screen." 

Further  problems  for  investigation  by  the  student  then  are: 

1.  The  investigation  of  the  effect  of  the  distortion  of  the  wave 
of  impressed  e.m.f.  on  the  constant  current,  with  other  trans- 
forming devices,  and  also  the  reverse  problem,  the  investigation 
of  the  effect  of  the  distortion  of  the  constant-current  wave,  as 
caused  by  an  arc,  on  the  system  of  transformation. 

2.  What  must  be  the  value,  x\,  of  the  reactance  of  a  wave 
screen,  to  reduce  the  wave-shape  distortion  of  the  secondary 
current  in  the  monocyclic  square  to  the  same  percentage  as  the 
distortion  of  the  impressed  e.m.f.  wave,  or  to  any  desired  per- 
centage, or  to  reduce  the  variation  of  the  constant  current  with 
the  load,   as  due  to  the  wave-shape  distortion,  below  a  given 
percentage? 

3.  Determination  of  efficiency  and  regulation  in  the  mono- 
cyclic  square  with  interposed  wave  screen,  Xi,  assuming  again 
3  per  cent,  loss  in  the  inductances,  1  per  cent,  loss  in  the  capacities 
and  choosing  z4  so  as  to  fill  given  conditions,  regarding  wave- 
shape distortion,  or  regulation,  or  efficiency,  etc. 


CHAPTER  XV 
CONSTANT-VOLTAGE  SERIES  OPERATION 

155.  Where  a  considerable  number  of  devices,  distributed  over 
a  large  area,  and  each  consuming  a  small  amount  of  power,  are  to 
be  operated  in  the  same  circuit,  low-voltage  supply — 110  or  220 
volts — usually  is  not  feasible,  due  to  the  distances,  and  high- 
voltage  distribution — 2300  volts — with  individual  step-down 
transformers  at  the  consuming  devices,  usually  is  uneconomical, 
due  to  the  small  power  consumption  of  each  device. 

In  such  a  case,  series  connection  of  the  devices  is  the  most  eco- 
nomical arrangement,  and  therefore  commonly  used. 

Such  for  instance  is  the  case  in  lighting  the  streets  of  a  city,  etc. 

Most  of  the  street  lighting  has  been  done  by  arc  lamps  operated 
on  constant-current  circuits,  and  as  the  universal  electric  power 
supply  today  is  at  constant  voltage,  transformation  from  constant 
voltage  to  constant  current  thus  is  of  importance,  and  has  been 
discussed  in  Chapter  XIV. 

The  constant-current  system  thus  is  used  in  this  case : 

(a)  Because  by  series  connection  of  the  consuming  devices,  as 
the  arc  lamps  in  street  lighting,  it  permits  the  use  of  a  sufficiently 
high  voltage  to  make  the  distribution  economical. 

(6)  The  dropping  volt-ampere  characteristic  of  the  arc  makes  it 
unstable  on  constant  voltage,  as  further  discussed  in  Chapters  II 
and  X,  and  a  constant-current  circuit  thus  is  used  to  secure  sta- 
bility of  operation  of  series  arc  circuits. 

The  condition  (6) ,  the  use  of  constant  current,  thus  applies  only 
where  the  consuming  devices  are  arcs,  and  ceases  to  be  pertinent 
when  the  consuming  devices  are  incandescent  lamps  or  other  con- 
stant-voltage devices. 

The  modern  incandescent  lamp,  however,  is  primarily  a  con- 
stant-voltage device,  that  is,  at  constant-voltage  supply,  the  life 
of  the  lamp  is  greater  than  at  constant-current  supply,  assuming 
the  same  percentage  fluctuation  from  constancy.  The  reason  is: 
a  variation  of  voltage  at  the  lamp  terminals,  by  p  per  cent.,  gives 
a  variation  of  current  of  about  0.6p  per  cent.,  and  thus  a  variation 

297 


298  ELECTRIC  CIRCUITS 

of  power  of  about  l.6p  per  cent.,  while  a  variation  of  current  in  the 

7? 

lamp,  by  p  per  cent.,  gives  a  variation  of  voltage  of  about  ~^  per 

cent.,  and  thus  a  variation  of  power  of  about  (1  +  7^)p  =  2.67p 

per  cent. 

Thus,  with  the  increasing  use  of  incandescent  lamps  for  street 
illumination,  series  operation  in  a  constant-voltage  circuit  be- 
comes of  increasing  importance. 

If  e  =  rated  voltage,  i  —  rated  current  of  lamp  or  other  con- 
suming device,  and  e0  =  supply  voltage,  n  =  —  lamps  can  be  op- 

€ 

erated  in  series  on  the  constant-voltage  supply  eQ.  If  now  one 
lamp  goes  out  by  the  filament  breaking,  all  the  lamps  of  the  series 
circuit  would  go  out,  if  eQ  is  small ;  if  e0  is  large,  an  arc  will  hold  in 
the  lamp  or  the  fixture,  and  more  or  less  destroy  the  circuit. 

Thus  in  series  connection,  especially  at  higher  supply  voltage, 
€Q,  some  shunt  protective  device  is  necessary  to  maintain  circuit 
in  case  of  one  of  the  consuming  devices  open-circuiting 

On  constant-current  supply,  a  short-circuiting  device,  such  as  a 
film  cutout,  takes  care  of  this.  With  series  connection  on  con- 
stant-voltage supply,  it  is  not  permissible,  however,  to  short- 
circuit  a  disabled  consuming  device,  as  this  would  increase  the 
voltage  on  the  other  devices.  Thus  the  shunt  protective  device 
in  the  constant- voltage  series  system  must  be  such,  that  in  case  of 
one  lamp  burning  out,  the  shunt  consumes  such  a  voltage  as  to 
maintain  the  voltage  on  the  other  devices  the  same  as  before.  A 
film  cutout,  with  another  lamp  in  series,  would  accomplish  this: 
if  a  lamp  burns  out,  its  shunting  film  cutout  punctures  and  puts 
the  second  lamp  in  circuit.  However,  in  general  such  arrange- 
ment is  too  complicated  for  use. 

As  practically  all  such  circuits  would  be  alternating-current 
circuits,  and  thus  alternating  currents  only  need  to  be  considered, 
the  question  arises,  whether  a  reactance  shunting  each  lamp 
would  not  give  the  desired  effect.  Suppose  each  lamp,  of  resist- 
ance, r,  is  shunted  by  a  reactance,  x,  which  is  sufficiently  large  not 
to  withdraw  too  much  current  from  the  lamp :  assuming  the  cur- 
rent shunted  by  x  is  20  per  cent,  of  the  current  in  the  lamp,  or  x 
=  5  r.  With  6.6  amp.  in  r,  x  thus  would  take  1.32  amp.,  and  the 
total,  or  line  current  would  be:  i  =  \/6.62  +  1.322  =  6.73  amp., 
thus  only  2  per  cent,  more  than  the  lamp  current.  If  now  a  lamp 


CONSTANT-VOLTAGE  SERIES  OPERATION      299 

burns  out,  the  total  current  flows  through  x,  instead  of  20  per  cent. 
only,  and  the  voltage  consumed  by  x  is  increased  fivefold  —  assum- 
ing x  as  constant  —  this  voltage,  however,  is  in  quadrature  with  the 
current,  thus  combines  vectorially  with  the  voltages  of  the  other 
consuming  devices,  which  are  practically  in  phase  with  the  cur- 
rent, and  the  question  then  arises,  whether,  and  under  what  con- 
ditions such  a  reactance  shunt  would  maintain  constant  voltage 
on  the  other  consuming  devices,  or,  what  amounts  to  the  same, 
constant  current  in  the  series  circuit. 

Such  a  reactance  shunting  the  consuming  device  could  at  the 
same  time  be  used  as  autotransformer  (compensator),  to  change 
the  current,  so  that  consuming  devices  of  different  current  re- 
quirements, as  lamps  of  various  sizes,  could  be  operated  in  series 
on  the  same  circuit,  from  constant-voltage  supply. 

156.  Let  n  lamps  of  voltage,  e\,  and  current,  i\t  thus  conductance 

r-S  a) 

be  connected  in  series  into  a  circuit  of  supply  voltage, 

e0  =  nei  (2) 

and  each  lamp  be  shunted  by  a  reactance  of  susceptance,  6. 

In  each  consuming  device,  comprising  lamp  and  reactance,  the 
admittance  thus  is,  vectorially, 

Fi  =  g  -  jb  (3) 

if,  then, 

/  =  current  in  the  series  circuit,  the  voltage  consumed  by 
the  device  comprising  lamp  and  reactance,  thus  is 

fl  =  71  =  J^J> 

in  a  consuming  device,  however,  in  which  the  lamp  is  burned  out, 
and  only  the  reactance  remains,  the  admittance  is 

Ft  *  -  jb  (5) 

hence,  the  voltage,  with  the  entire  current,  /,  passing  through  the 
admittance,  F2, 


If,  then,  of  the  n  series  lamps,  the  fraction,  p,  is  burned  out, 
leaving  n(l  —  p)  operative  lamps,  it  is: 


300  ELECTRIC  CIRCUITS 

voltage  consumed  by  operative  devices: 


voltage  consumed  by  devices  with  burned-out  lamps: 


thus,  total  circuit  voltage: 

e0  =  n(l  -  p)  E!  +  npEz  (7) 


or, 

_  nl(b  +  jpg)  ,  . 

€Q   —   —77  --  rrr—  (8) 

b(g  -  jb) 
or,  absolute, 


.. 

where 

?/  =  \/g2  -\-  b2  =  admittance  of  operative  device,  absolute,     (10) 

hence, 


is  the  current  in  the  circuit,  and  the  current  in  the  lamps  thus  is 

»'i  =  §*'  (12) 

hence, 


for 

p  =  0,  or  all  devices  operative  (ufull-load,"  as  we  may  say),  it 


is 


for 

p  =  1,  or  all  lamps  out  ("no-load"),  it  is 


CONSTANT-VOLTAGE  SERIES  OPERATION       301 


1\  ••= 


\u/ 

(14) 


ny 


thus  smaller  than  at  full-load. 

As  seen  from  equation  (13),  the  current  steadily  decreases,  from 
p  =  0  or  full-load,  to  p  =  1  or  no-load,  and  no  value  of  shunted 
reactance,  6,  exists,  which  maintains  constant  current.  With  de- 
creasing load,  the  current,  ii,  decreases  the  slower,  the  higher  6  is, 
that  is,  the  more  current  is  shunted  by  the  reactive  susceptance,  6, 
and  the  poorer  therefore  the  power-factor  is. 

Thus  shunted  constant  reactance  can  not  give  constant-voltage 
regulation. 

However,  with  b  =  0.2  gr,  at  no-load  the  shunted  reactance 
would  get  five  times  as  much  current  as  at  load,  and  thus  have  five 
times  as  high  a  voltage  at  its  terminals. 

The  latter,  however,  is  not  feasible,  except  by  making  the 
reactance  abnormally  large  and  therefore  uneconomical. 

In  general,  long  before  five  times  normal  voltage  is  reached, 
magnetic  saturation  will  have  occurred,  and  the  reactance  thereby 
decreased,  that  is,  the  susceptance,  6,  increased,  as  more  fully  dis- 
cussed in  Chapter  VIII. 

This  actual  condition  would  correspond  to  a  value,  61,  of  the 
shunted  susceptance  when  shunted  by  the  lamp,  and  a  different, 
higher  value,  62,  of  the  shunted  susceptance  when  the  lamp  is 
burned  out. 

The  question  then  arises,  whether  such  values  of  61  and  62  can  be 
found,  as  to  give  voltage  regulation.  The  increase  of  62  over  61 
naturally  depends  on  the  degree  of  magnetic  saturation  in  the  re- 
actance, that  is,  on  the  value  of  magnetic  density  chosen,  and 
thus  can  be  made  anything,  depending  on  the  design. 

167.  Let  then,  as  heretofore, 


EQ  =  eo  =  constant-supply  voltage. 

/    =  current  in  series  circuit. 

n    =  number  of  consuming  devices  (lamps)  in  series. 

p    —  fraction  of  burned-out  lamps. 

g    =  conductance  of  lamp. 


(15) 


(16) 


302  ELECTRIC  CIRCUITS 

and  let 

61  =  shunted  susceptance  with  the  lamp  in  circuit, 

that  is,  exciting  susceptance  of  reactor  or  auto- 
transformer,  and 

y     =  \/g2  -f-  bi2  =  admittance  of  complete  consuming 
device. 

62  =  shunted  susceptance  with  the  lamp  burned  out 

and  let 

c  =  — L  =  exciting  current  as  fraction  of  load 

^  current:  c  <  1.  ,-.-, 

a  =  ~-  =  saturation    factor  of   reactor    or 

autotransformer:  a  >  1. 
it  is,  then: 

voltage  of  lamp  and  reactor: 

#1  =      _5  >b  (18) 

voltage  of  reactor  with  lamp  burned  out: 


thus, 

with  pn   lamps  burned  out,  and  (1  —  p)n  lamps  burning,  it  is 
total  voltage, 

—   fv\  ( "I        fY\\  77*       ]     /v)/x»    ~FF  ( 'OOA 

=  n  T  [  1  ~  P  +  j  P.' 

substituting  (17), 


or» 

nl  1  —  p(l  —  ac)  +  jap 
e°  =:  7  '  1  _jc          ~> 

hence,  absolute, 

2/ 
since, 

2/  =  0Vl  +  c2 
thus,  the  current  in  the  series  circuit, 

eoy 


(24) 


CONSTANT-VOLTAGE  SERIES  OPERATION       303 
158.  For,  p  =  0,  or  full-load,  it  is 


e°y  com 

—  (25) 


i  =  7°  (26) 


thus, 


The  same  value  of  iQ  as  at  full-load  is  reached  again  for  the 
value  p  =  po,  where  the  square  root  in  (24)  becomes  one,  that  is, 

[l-po(l-ac)]2+a2p02  =  1, 
hence, 

2(1  -  ac) 
P°  ~  a2  +  (1  -  ac)2 

for,  p  =  1,  or  no-load,  it  is 

V   =  ^ 

aVl  +  c2 


(28) 


The  current  is  a  maximum,  i  =  im,  for  the  value  of  p  =  pm,  given 
by 


or,  from  (26), 

^{[1  -p(l  -ac)]«+aVI  =0, 
this  gives 


2' 

as  was  to  be  expected. 

Substituting  (29)  into  (26) ,  gives  as  the  value  of  maximum  cur- 
rent 

/  /I       sift\     9 

(30) 


and  the  regulation  5,  that  is,  the  excess  of  maximum  current  over 
full-load  current,  as  fraction  of  the  latter,  thus  is 


to 


304 


ELECTRIC  CIRCUITS 


If  q  is  small  (31)  resolved  by  the  binomial,  gives 

2 


(32) 


As  seen,  with  the  shunted  susceptance  increased  by  saturation 
at  open  circuit,  the  current  and  thus  lamp  voltage  are  approxi- 
mately constant  over  a  range  of  p.  That  is,  with  decreasing  load, 
from  full-load  p  =  0,  the  current  i,  and  proportional  thereto  the 

lamp  voltage  increases  from  io  to  a  maximum  value  im,  at  p  »?r, 

2 

then  decreases  again,  to  i0  at  p  —  pQ,  and  decreases  further,  to  i\ 
at  no-load,  p  =  I. 

Thus,  there  exists  a  regulating  range  from  p=  Q  to  p  a  little 
above  PQ,  where  the  current  is  approximately  constant. 

Instance: 


Saturation:   a    = 

1.5 

1.5 

1.5 

2.0 

2.0 

2.0 

2.5 

2.5 

2.5 

Excitation  :    c     = 

0.1 

0.2 

0.3 

0.1 

0.2 

0.3 

0.1 

0.2 

0.3 

Regulation:  q    = 
Range:           p0  = 

0.147 
0.573 

0.103 
0.510 

0.067 
0.432 

0.077 
0.345 

0.044 
0.275 

0.020 
0.192 

0.044 
0.220 

0.020 
0.154 

0.005 
0.079 

IV 


-50 


10  20 


40  50 


CO 


90  100% 


FIG.  126. 
As  illustrations  are  shown,  in  Fig.  126,  the  regulation  curves,  — , 

from  equation  (26),  for: 

a  =  1.5  c  =  0.2  Curve  I 

=  2.5  =0.1                       II 

=  2.0  =  0.3                      III 


CONSTANT-VOLTAGE  SERIES  OPERATION       305 

159.  By  the  preceding  equations,  it  is  possible  now  to  calculate 
the  values  of  exciting  susceptance  61,  and  saturation  bz,  required 
by  the  shunting  reactors  to  give  desired  values  of  regulation  with- 
in a  given  range. 

From  (32)  follows: 

c  =-a-  Vlq  (33) 

Substituting  (33)  into  (27)  gives: 

(34) 

c  == — 

2\/<Z 

From  chosen  values  of  q  and  p0,  a  and  c  thus  can  be  calculated, 
from  a  and  c  and  the  conductance  g  of  the  consuming  device,  61, 
62,  i,  etc.,  follow. 

Instance: 

n  =  100  lamps  of  ii  =  6  amp.  and  e\  —  50  volts,  are  to  be  oper- 
ated in  series  on  constant-voltage  supply,  with  negligible  line  re- 
sistance and  reactance.     The  regulation  shall  be  within  4  per  cent, 
in  a  range  of  30  per  cent.     That  is,  q  =  0.04  and  po  =  0.30. 
It  thus  is : 

t!    =    6 

ci  =  50 


n  =  100 
q  =  0.04 

PQ   =  0.30 

From  (34)  follows: 

a  =  1.75 
c  =  0.287 

Hence  by  (17): 

61  =  0.0345 

62  =  0.0685 
and  by  (16): 

y  =  0.1248 
by  (2): 

eo  =  5000  volts 

20 


306 


ELECTRIC  CIRCUITS 


and  by  (25) : 

to  =  6.24  amp. 
thus,  by  (26)  • 


i  = 


6.24 


VI  -  P  +  3.31  p* 

Fig.  127  shows,  as  curve  I,  the  values  of  q  =  — 
cent.,  that  is,  the  regulation,  with  p  as  abscissas. 


(35) 
1,  in  per 


10     12    14    16     18 


22    24     26     28     30    82     84     36     38     40% 


FIG.  127. 

160.  In  general,  the  resistance  and  reactance  of  the  circuit  or 
line  is  not  negligible,  as  assumed  in  the  preceding,  and  the  re- 
actors, especially  if  used  at  the  same  time  as  autotransformers, 
contain  a  leakage  reactance,  which  acts  as  a  series  reactance  in  the 
circuit,  and  the  lamp  circuit  of  conductance  g  also  may  contain  a 
small  series  reactance. 
Let  then : 

r0  =  line  resistance ; 
XQ  =  line  reactance; 
x    =  series  or  leakage  reactance  per  autotransformer 
or  consumption  device. 

The  most  convenient  way  is  to  represent  r0,  x0  and  x  by  their 
equivalent  in  lamps  or  reactors.  The  admittance  of  each  con- 
sumption device,  comprising  lamp  and  reactor  or  autotrans- 
former, is 


CONSTANT-VOLTAGE  SERIES  OPERATION       307 

Yi  =  g  -j6i  =  0(1  -  jc), 
thus  the  impedance, 

Zl  =  7i  =  0(1  -  jc)  =  0(1  +  c2)' 
and  by  (23), 


irv/i  +  c2' 

If,  then,  we  add  to  the  resistance  r0  a  part  cr0  of  the  reactance, 
we  get  an  impedance, 


which  has  the  same  phase  angle  as  Z\t  and  thus  can  be  expressed 
as  a  multiple  of  Zi, 

Z  =  mZi, 
where 

ni  =  f-  =  r0yVl  +  c2  (36) 

^i 

thus  is  the  "lamp  equivalent"  of  the  line  resistance  ro  plus  the 
part  cr0  of  the  reactance. 
This  leaves  the  reactance, 

Xi  =  XQ  +  n(l  —  p)x  —  cr0, 
and  as  the  reactance  of  a  reactor  without  lamp  is 

12  •  £• 

the  reactance  x\  can  be  expressed  as  multiple  of  Z2, 

Xi  =  nzx2 


where 

b2  [x0  +  n(l  —  p)x  —  cr0]  (37) 


thus  is  the  "lamp  equivalent"  of  the  line  reactance  XQ  and  leakage 
reactances  x\  in  burned-out  lamps. 

Thus  the  addition  of  the  line  impedance  ro  +  jxQ,  and  the 
leakage  reactances  x,  is  represented  by  HI  lamps  with  reactors, 
and  n2  burned-out  lamps,  or  a  total  of  n\  +  n2  lamps. 

Thus  the  circuit  can  carry  ni  —  (HI  +  n2)  lamps,  and  its  regula- 

tion curve  starts  at  the  point  p  =  —  and  ends  at  p  =  1  ---  -  — 
of  the  complete  regulation  curve. 


308  ELECTRIC  CIRCUITS 

However,  in  this  case,  the  full-load  current,  for  p  =  — ,  would 

already  be  slightly  higher  than  in  a  circuit  without  line  impedance, 
and  all  the  current  values  would  thus  have  to  be  proportionally 
reduced. 

Instance: 

In  the  case 

a    =  2.0 

c    =  0.3 
given  as  curve  III  of  Fig.  126  let: 

n   =     100 

g    =  0.12 

r0  =     50 

x0  =  0.5 
hence, 

6j  =  Cg  =  0.036 

=   0.125 

0.06, 

thus, 

Wi  =  rQy  Vl  +  c2  =  6.54 

n2  =  62  [x0  +  n  (I  —  p)  x  —  cr0]  =  7.0 

n\  -f-  ft2  =  13.54. 

Thus,  the  regulation  curve  starts  at  p  =  --  =  0.07  of  curve 

III,  Fig.  126,  and  ends  at  p  =  1  -  —  =  0.935  of  this  curve. 
For  p  =  0.07  it  is,  by  equation  (26), 

4  =  1.017, 

i>o 

thus,  all  values  of  curve  III,  Fig.  126,  are  reduced  by  dividing 
with  1.017,  and  then  plotted  from  p  =  0.07  on,  and  then  give 
the  regulation  curve  inclusive  line  resistance  shown  as  curve  IV. 

As  seen,  the  regulation  range  is  reduced,  but  the  regulation 
greatly  improved  by  the  line  impedance.  This  is  done  essen- 
tially by  the  line  reactance  and  leakage  reactance,  but  not  by 
the  resistance. 

161.  Instead  of  approximating  the  effect  of  line  impedance 
and  leakage  reactance  by  equivalent  lamps  and  reactors,  it  can  be 
directly  calculated,  as  follows: 


CONSTANT-VOLTAGE  SERIES  OPERATION      309 

Lot 

r0  =  line  resistance 
XQ  —  line  reactance 
x    =  leakage  or  series  reactance  per 
autotransformer 

the  other  symbols  being  the  same  as  (15),  (16)  and  (17). 

It  is  then : 
voltage  consumed  by  line  resistance  r0: 

XI 

voltage  consumed  by  line  reactance  XQ  : 

jxol 

voltage  consumed  by  leakage  reactances  x  of  the  n(l  —  p)  lamp 
devices : 

jxn(l 


thus,  total  circuit  voltage: 

substituting  the  abbreviation, 

n 

(40) 

h3  =  xg 

and  substituting  (17)  and  (40)  into  (39),  gives 
Inil  - 


e°  =  " 


_ 


7!  [r^  +  ftJ  ^[^c+  i-  +  i-*-  a  -  P)*.]  ) 


hence,  absolute, 


hence,  the  current, 

i  = 


'(43) 


310  ELECTRIC  CIRCUITS 

for  p  =  0  (42)  and  (43)  gives  the  full-load  current  and  voltage, 


^  +  *•  +  "'        (44) 

where  (12) 

to  =  ti  -  (45) 

is  the  full-load  line  current,  for  i\  =  full-load  lamp  current. 
162.  Let,  in  the  instance  paragraph  159  and  Fig.  126; 

r0  =  50 

XQ   =    75 

x    =  0.5 

the  other  constants  remaining  the  same  as  in  paragraph  159, 
that  is: 

ti  =  6 

n   =  100 

g    =  0.12 

61  =  0.0345 

62  =  0.0685 
y    =  0.1248 
a    =  1.75 

c    ==  0.287 

It  is  then  (40), 

/ii  =  0.06 
/i2  =  0.09 
ht  =  0.06 

hence,  by  (45), 

to  =  1.04  X  6  =  6.24  amp. 

by  (44),  __ 

e0  =  5200  V(a923_+_0.06)2  +  (0.264  +  0.09  +  0.06)  2 

=  5200  \/0.9832  +  4142 

=  5200  \/1.137 

=  5200  X  1.066 
eo  =  5540  volts 

and  by  (43), 

.  =  _  6.65  _ 
"  V(0.983  -  0.923  p)2  +  (0.414  +  1.426  p)2 


CONSTANT-VOLTAGE  SERIES  OPERATION       311 

= 6.65 

"  \/1.137  -  0.634  p  +  2.885  p2 

i  -        6'24  (46) 

•N/l  -  0.558  p  +  2.54  p2 

Fig.    127  shows,   as   curve  II,  the  values   of  ^-^  —  I  from 

equation  (46),  that  is,  the  regulation,  as  modified  by  line  imped- 
ance and  leakage  reactance,  with  p  as  abscissae. 

The  regulating  range,  p0,  of  equation  (46)  is  given  by 

1  -  0.558  po  +  2.54  p02  =  1, 
hence, 

po  =  0.22. 

Thus  the  regulation  range  is  reduced  by  the  line  impedance  and 
leakage  reactance,  from  30  per  cent,  to  22  per  cent. 
The  maximum  value  of  current,  im,  occurs  at 

P. -f- 0.11 

and  is  given  by  substitution  into  (46),  as, 

fcfi-  1-015, 

or, 

q  =  0.015. 

That  is,  the  regulation  is  improved,  by  the  line  and  leakage 
reactance,  from  q  =  4  per  cent,  to  q  =  1.5  per  cent,  as  seen  in 
Fig.  127. 

163.  In  paragraph  161  and  the  preceding,  the  shunted  react- 
ances, bi  and  62,  have  been  assumed  as  constant  and  independent 
of  p.  However,  with  the  change  of  p,  the  wave-shape  distortion 
between  current  and  voltage  changes,  as  with  increasing  p,  more 
and  more  saturated  reactors  are  thrown  into  the  circuit  and  dis- 
tort the  current  wave. 

As  61  is  shunted  by  g,  and  carries  a  small  part  of  the  current 
only,  and  g  is  non-inductive,  the  change  of  wave  shape  in  61 
will  be  less,  and  as  &i  carries  only  a  part  of  the  current,  the 
effect  of  the  change  of  wave  shape  in  61  thus  is  practically  neg- 
ligible, so  that  bi  can  be  assumed  as  constant  and  independent  of  p. 

bz,  however,  carries  the  total  current,  at  fairly  high  saturation, 
and  thus  exerts  a  great  distorting  effect. 

At  and  near  full-load,  with  all  or  nearly  all  conductances,  g,  in 


312  ELECTRIC  CIRCUITS 

circuit,  the  entire  circuit  is  practically  non-inductive,  that  is,  the 
current  has  the  same  wave  shape  as  the  voltage.  Assuming  a 
sine  wave  of  impressed  voltage,  60,  the  current,  i,  at  and  near  full- 
load  thus  is  practically  a  sine  wave,  and  the  shunting  reactance, 
bz,  thus  has  the  value  corresponding  to  a  sine  wave  of  current 
traversing  it,  that  is,  the  value  denoted  as  "constant-current 
reactance,"  xc,  in  Chapter  VIII. 

At  no-load,  with  all  or  nearly  all  conductances,  g,  open-circuited, 
the  entire  circuit  consists  of  a  series  of  n  reactive  susceptances,  62. 
If,  then,  the  impressed  voltage,  e0,  is  a  sine  wave,  each  susceptance, 
62,  receives  1/n  of  the  impressed  voltage,  thus  also  a  sine  wave. 
That  is,  at  and  near  no-load,  the  shunted  reactance,  62,  has  the 
value  corresponding  to  an  impressed  sine  wave  of  voltage,  that 
is,  the  value  denoted  as  "constant-potential  reactance,"  xp,  in 
Chapter  VIII. 

xc,  however,  is  materially  larger  than  xp,  and  the  shunting  re- 
actance thus  decreases,  that  is,  the  shunting  susceptance,  62,  in- 
creases from  full-load  to  no-load,  or  with  increasing  p. 

Due  to  the  changing  wave-shape  distortion,  62  thus  is  not  con- 
stant, but  increases  with  increasing  p,  thus  can  be  denoted  by 

62  =  6o(l  +  sp)  (47) 

this  gives 

' 


Substituting  (48)  into  (43)  gives,  as  the  equation  of  current, 
allowing  for  the  change  of  wave-shape  distortion, 


(49) 


Assume,  in  the  instance  paragraphs  159  and  161,  and  Fig.  127, 
that  the  shunted  susceptance,  62,  increases  from  full-load  to  no- 
load  by  40  per  cent.  That  is, 

s  =  0.4; 
it  is,  then, 


a  = 


1  +  0.4  p 

Assuming  now,  that  at  the  end  of  the  regulating  range, 
p  =  PQ  =  0.22, 


CONSTANT-VOLTAGE  SERIES  OPERATION       313 

o  has  the  same  value  as  before, 

a  =  1.75, 
this  gives 

1.75  = 


1  +  0.4  X  0.22 

a  =  1.90 
and 

1.9 


(50) 


1  +  0.4  p 
Substituting  now  the  numerical  values  in  equation  (49),  gives 

.  =  6.65 

"  \/(0.983  -  0.923  p)2  +  (0.414  +  (a  -  0.324)p)2 

= 6.24 == 

~  V[0.928  -  0.866  p]2  +  [0.388  +  (0.938  a  -  0.304)  p]2 

Fig.  127  shows,  as  curve  III,  the  values  of  ^r  from  equation 

(51),  that  is,  the  regulation  as  modified  by  the  changing  wave 
shape  caused  by  the  saturated  reactance. 

The  maximum  value  of  current,  im,  occurs  at  pm  =  ^  =  0.11, 
and  is  given  by  substitution  into  (50)  and  (51),  as, 

a  =  1.82 


, 

that  is, 

q  =  0.011 

thus,  the  regulation  is  still  further  improved,  by  changing  wave 
shape,  to  1.1  per  cent. 


CHAPTER  XVI 
LOAD    BALANCE   OF   POLYPHASE   SYSTEMS 

163.  The  total  flow  of  power  of  a  balanced  symmetrical  poly- 
phase system  is  constant.  That  is,  the  sum  of  the  instantaneous 
values  of  power  of  all  the  phases  is  constant  throughout  the  cycle. 
In  the  single-phase  system,  however,  or  in  a  polyphase  system  with 
unbalanced  load,  that  is,  a  system  in  which  the  different  phases 
are  unequally  loaded,  the  total  flow  of  power  is  pulsating,  with 
double  frequency.  To  balance  an  unbalanced  polyphase  system 
thus  requires  a  storage  of  energy,  hence  can  not  be  done  by 
any  method  of  connection  or  transformation.  Thus  mechanical 
momentum  acts  as  energy-storing  device  in  the  use  as  phase  bal- 
ancer, of  the  induction  or  the  synchronous  machine.  Electrically, 
energy  is  stored  by  inductance  and  by  capacity.  The  question 
then  arises,  whether  by  the  use  of  a  reactor,  or  a  condenser,  con- 
nected to  a  suitable  phase  of  the  system,  an  unequally  loaded 
polyphase  system  can  be  balanced,  so  as  to  give  constant  power 
during  the  cycle. 

In  interlinked  polyphase  circuits,  such  as  the  three-phase  sys- 
tem, with  unbalanced  load  carried  over  lines  of  appreciable  im- 
pedance, the  voltages  of  the  three  phases  become  unequal.  This 
makes  voltage  regulation  more  complicated  than  in  a  balanced 
system.  A  great  unbalancing  of  the  load,  such  as  produced  by 
operating  a  heavy  single-phase  load,  as  a  single-phase  railway  or 
electric  furnace,  greatly  reduces  the  power  capacity  of  lines,  trans- 
formers and  generators.  Unbalanced  load  on  the  generators 
causes  a  pulsating  armature  reaction:  at  single-phase  load,  the 
armature  reaction  pulsates  between  more  than  twice  the  average 
value,  and  a  small  reversed  value,  between  F(cos  a  +  1)  and 
F(cos  a  —  1),  where  cos  a  is  the  power-factor  of  the  single-phase 
load.  Especially  in  alternators  of  very  high  armature  reaction, 
as  modern  steam-turbine  alternators,  a  pulsation  of  the  armature 
reaction  is  very  objectionable.  It  causes  a  pulsation  of  the  field 
flux,  leading  to  excessive  eddy-current  losses  and  consequent  re- 
duction of  the  output.  The  use  of  a  squirrel-cage  winding  in  the 

314 


LOAD  BALANCE  OF  POLYPHASE  SYSTEMS      315 

field  pole  faces  of  the  single-phase  alternator  reduces  the  pulsation 
of  the  field  flux,  but  also  increases  the  momentary  short-circuit 
stresses. 

Thus,  it  is  of  interest  to  study  the  question  of  balancing  unbal- 
anced polyphase  circuits  by  stationary  energy-storing  devices,  as 
reactor  or  condenser. 

164.  Let  a  voltage, 

e  =  E  cos  </>  (1) 

be  impressed  upon  a  non-inductive  load,  giving  the  current 

i  =  I  cos  0  (2) 

The  power  then  is 

p  =  ei  =  El  cos2  </> 
•pj 

=  ^  (1  +  cos  2  *) 

=  Q  +  Q  cos  2  </>  (3) 

where 

«-¥  <« 

that  is, 

in  a  non-inductive  single-phase  circuit,  the  power  consists  of  a 

constant  component, 

«-¥•        '     •••  * 

and  an  alternating  component, 

EI 

Q    =   -y  COS  2  0, 

of  twice  the  frequency  of  the  supply  voltage,  and  a  maximum 
value  equal  to  that  of  the  constant  component.     The  instantane- 
ous power  thus  pulsates  between  zero  and  2  Q,  by  equation  (3). 
If  the  circuit  is  inductive,  of  lag  angle  a,  the  current  is 

i  =  I  cos  (<£  —  a)  (5) 

and  the  instantaneous  power  thus, 

p  =  EI  cos  </>  cos  (<£  —  a) 


cos  a  +  cos  (2  0  —  a) 


=  P  +  Q  cos  (2  0  -  «), 
thus  consists  of  a  constant  component, 


ETT 

P  =  -jr-  cos  a  =  Q  cos  a  (7) 

a 


316  ELECTRIC  CIRCUITS 

and  an  alternating  component, 

Q  cos  (2  0  -  a)  ; 

it  thus  pulsates  between  a  small  negative  and  a  large  positive 
value,  P  -  Q  and  P  +  Q. 

If  the  circuit  is  completely  inductive,  that  is,  the  current  lags 

90°  or  ~  behind  the  voltage,  the  current  is 


(8) 
and  the  instantaneous  power  thus, 

p  =  El  cos  <f>  cos(0  —  ^j 


—  sin  2 
4 


Thus,  the  power  comprises  only  an  alternating  component,  but 
no  continuous  component;  in  other  words,  no  power  is  consumed, 
but  the  power  surges  or  alternates  between  +Q  and  —  Q,  that  is, 
power  is  stored  and  then  again  returned  to  the  circuit. 

If  the  circuit  is  closed  by  a  capacity,  C,  the  current  leads  the 

impressed  voltage  by  ~,  thus  is 


(10) 

and  the  instantaneous  power  thus, 

p  =  El  cos  0  cos  (  0  +  ^J 

(11) 


thus,   comprises   only   an   alternating   component,   surging  be- 
tween —  Q  and  +Q,  with  double  frequency. 

The  power  consumed  by  a  condenser,  equation  (11),  is  opposite 
in  sign  and  thus  in  direction,  from  that  consumed  by  a  reactor  (9), 

Q  cos(2  0  +  £)   =  -  Q  cos(2  0  -  £)  • 
\  2tl  \  £' 

165.  If  a  number  of  voltages, 

d  =  Ei  cos  (0  —  7i)  (12) 

1  "Engineering  Mathematics,"  Chapter  III,  paragraphs  66  to  75. 


LOAD  BALANCE  OF  POLYPHASE  SYSTEMS      317 

of  a  polyphase  system,  produce  currents, 

ii   =  Ii  cos  (</>  -  7»  -  «.)  (13) 

the  instantaneous  power  of  each  voltage  e<  is 

Pi  =  edi 

=  Q»{cos  cti  +  cos  (2  </>  -  2  7<  -  «»)}  (14) 

and  the  total  instantaneous  power  of  the  system  thus  is 

P    =  Sp< 

=  S<3<  cos  <*i  +  SQ»  cos  (2  </>  -  2  7i  -  «<) 
=  P  +  Q  cos  (2  0  -  a)  (15) 

where 

P    =  2QiCOSa:<  (16) 

is  the  total  effective  power  of  the  system,  and 

Q  =  2Q»  cos  (2  0  -  2  7.  -  «0  (17) 

is  the  total  resultant  alternating  component  of  power,  or  the 
resultant  power  pulsation  of  the  system. 

Thus,  the  power  of  the  polyphase  system  pulsates,  with  double 
frequency,  between  P  —  Q  and  P  +  Q. 

In  this  case,  P  may  be  greater  than  Q,  and  frequently  is,  and  the 
power  thus  pulsates  between  two  positive  values,  while  in  the 
single-phase  circuit  (6)  it  pulsated  between  positive  and  negative 
value. 

It  thus  can  be  seen,  that  in  any  system,  polyphase  or  single- 
phase,  with  any  kind  of  load,  the  total  instantaneous  power  of  the 
system  can  be  expressed, 

p  =  p  +  Q  cos  (2  <f>  -  a)  (18) 

where  P  is  the  constant  component  of  power,  and  Q  the  amplitude 
of  the  double-frequency  alternating  component  of  power,  and  Q 
may  be  larger  or  smaller  than  P. 

It  must  be  noted,  that  Q  is  not  the  total  reactive  power  of  the 
system — which  would  have  to  be  considered,  for  instance,  in 
power-factor  compensation  etc. — but  Q  is  the  vector  resultant 
of  the  reactive  powers  of  the  individual  circuits,  while  the  total 
reactive  power  of  the  system  is  the  algebraic  sum  of  the  individual 
reactive  powers  (see  "Theory  and  Calculation  of  Alternating- 
current  Phenomena,"  Chapter  XVI). 

Thus,  for  instance,  in  a  system  of  balanced  load,  even  if  the 
load  is  reactive,  Q  =  0.  Thus,  Q  is  the  unbalanced  reactive 


318  ELECTRIC  CIRCUITS 

power  of  the  system,  and  does  not  include  the  reactive  power, 
which  is  balanced  between  the  phases  and  thereby  gives  zero 
as  vector  resultant. 

166.  The  expression  of  the  power  of  a  polyphase  system  of  gen- 
eral unbalanced  load  is  by  (15) 

p  =  P  +  Q  cos  (2  0  -  a)  (19) 

this  also  is  the  expression  of  power  of  the  single-phase  load  of 
lag  angle  a,  of  the  impressed  voltage  and  current, 

e  =  E  cos  0 


i  =  I  cos  (0-  ' 


where,  from  (20), 

P  =  Q  sin  a 
El 


(21) 


2 

while  in  the  general  case  (19)  P  and  Q  may  have  any  values. 
Suppose  now  we  select  from  the  polyphase  system  a  voltage, 

e'  =  Er  cos  (0  -  |8)  (22) 

and  load  it  with  an  inductive  load  of  zero  power-factor, 

i'-/' COB  (*-/J-|)  (23) 

Er 
that  is,  we  connect  a  reactor  of  x  =  -y>  into  the  phase  e1 '. 

The  power  of  (22)  (23)  then  is 

p'  =  Q'cos(2  0-2/3-?)  (24) 

where 

Q'  =  ^  (25) 

and  the  total  power  of  the  system,  comprising  (19)  and  (25),  thus 
is 

Po  =  P  +  Pf 

=  P  +  Q  cos  (2  0  -  a)  +  Q'  cos  (2  0  -  2  0  -  ?) 

and  this  would  become  constant,  and  the  double-frequency  term 
eliminated,  that  is,  the  system  would  be  balanced,  if  Q'  and  0  are 
chosen  so  that 

Q  cos  (2  0  -  a)  +  Q'  cos  (2  0  -  2  0  -  ?)  =  0          (26) 

\         -  ^u/ 


LOAD  BALANCE  OF  POLYPHASE  SYSTEMS      319 

hence, 

Q'  =  Q  (27) 

2  0  -  2  0  -  I  =  2  <£  -  a  -  IT 
or, 


r-  (29) 

-?-s 

thus, 

«'-tfcos[*-(f  +  5]  (31) 

is  the  voltage,  which,  impressed  upon  a  reactor  of  reactance, 

E'2 
x  =  fg  (30) 

balances  the  power, 

p  =  P  +  Q  cos  (2  0  -  a)  (24) 

of  an  unbalanced  polyphase  system.     That  is, 

e'  =  E'  cos  [*-(!  +  ?)]  (31) 

impressed  upon  the  reactance,  x,  gives  the  current, 


and  thus  the  power, 


=  -  Q  cos  (2  0  -  a)  (33) 

and  this  reactive  power,  pf,  added  to  the  unbalanced  polyphase 
power,  p,  gives  the  balanced  power, 

p  =  p  +  p' 
=  P. 

167.  Comparing  (31)  with  (20)  or  (24),  it  follows: 
The  unbalanced  load  of  a  single-phase  voltage, 

e  =  E  cos  0, 


320  ELECTRIC  CIRCUITS 

of  lag  angle,  a,  or  in  general,  the  unbalanced  load  of  a  polyphase 
system  with  the  resultant  instantaneous  power  of  lag  angle,  a, 

p  =  P  +  Q  cos  (2  0  -  a) 

can  be  balanced  by  a  wattless  reactive  load,  p',  having  the  same 
volt-amperes,  Q',  as  the  alternating  component,  Q,  of  the  unbal- 
anced load,  and  having  a  phase  of  voltage  lagging  by 

«+5 

2  +4 

or  by  45°  plus  half  the  lag  angle,  a,  of  the  unbalanced  load  or  un- 
balanced single-phase  current. 

Just  as  the  unbalanced  polyphase  load,  p,  (24)  may  be  single- 
phase  load  on  one  phase,  or  the  vector  resultant  of  the  loads  on 
different  phases,  so  the  wattless  reactive  compensating  volt- 
amperes  (33)  may  be  due  to  a  single  reactor  connected  into  the 
compensating  voltage,  e' ',  (31),  or  may  be  the  vector  resultant  of 
several  voltages,  e'i,  loaded  by  reactances,  x\,  so  that  their 
vector  resultant  is  pf  (33). 

If  a  capacity  is  used  for  energy  storage  in  balancing  unbalanced 
load  (24),  the  compensating  voltage  (22), 

ef  =  Ef  cos  (0  -  0), 
impressed  upon  the  capacity  gives  the  reactive  leading  current, 

i'  =7'cos(0-/3+!)  (34) 

hence  the  compensating  reactive  power, 

p'  =E'I'  cos  (2  0_20  +  |)  (35) 

and  therefrom,  by  the  same  reasoning  as  before, 

a        3  TT  f     . 

13  =+ 


(37) 


That  is,  when  using  a  capacity  for  balancing  the  load,  the  com- 
pensating voltage,  e',  has  the  phase, 


LOAD  BALANCE  OF  POLYPHASE  SYSTEMS      321 

or,  what  is  the  same  as  regards  to  the  power  expression, 

a       TT 
2  "!' 
thus  lags  by  half  the  phase  angle,  a,  minus  45°  (or  plus  135°). 

168.  As  instance  may  be  considered  a  quarter-phase  system 
with  one  phase  loaded. 
Let 

e\  —  E  cos  <£ 


=  E 


(38) 


be  the  two  phase-voltages  of  the  quarter-phase  system. 

Let  the  first  phase,  e\,  be  loaded  by  a  current  lagging  by  phase 
angle,  a, 

ii  =  I  cos   (0  -  a)  (39) 

while  the  second  phase,  e^,  is  not  loaded. 
The  power  then  is 

P  =  eiii 

=  4r!cos  a  +  cos(2  <£  ~  «)}  (4°) 

& 

and  is  compensated  or  balanced  by  a  reactance  connected  to  a 
compensating  phase, 

e'  =  E'  cos  (</>-  0)  (41) 

and  consuming  the  reactive  current, 

(42) 

where  the  —  ~  represents  inductive  reactance,  the  +  ~  capacity 

reactance. 

The  compensating  reactive  power  then  is 


p'  =  e'i' 


P' T' 

(43) 


and  this  becomes  equal  to 
for 


El 

-  -       cos  (2  0  -  a), 


ET  =  El 
ft  =  I  +  1 


(44) 

ir  i  ^  4 

21 


322  ELECTRIC  CIRCUITS 

and  the  compensating  circuit  thus  is 


-  -  -  - 

it  is,  then, 

p'  =  e'i' 

=  E'I'  cos  (2  </>  -  a  +  TT) 
=  —El  cos   (2   0  —  a) 

hence, 

Po  =  p  +  Pf 

El 

=  -y  cos  a, 

for 

a  =  0,  or  non-inductive  load,  it  is 


(45) 


hence,  if  we  choose, 

TfT/      TTTf 

rj      ^    /t/ 

hence, 


it  is 

e'  =  6j  ±  62  (46) 

that  is,  connecting  the  two  phases  in  series,  gives  the  compensat- 
ing voltage  for  non-inductive  load.  Or: 

"Non-inductive  single-phase  load,  on  one  phase  of  a  quarter- 
phase  system,  can  be  balanced  by  connecting  a  reactance  across 
both  phases  in  series,  of  such  value  as  to  consume  a  current  equal 
to  the  single-phase  load  current  divided  by  \/2,  that  is,  having  the 
same  volt-ampere  as  the  single-phase  load." 

169.  In  the  general  case  of  inductive  load  of  power-factor,  a, 
the  compensating  voltage  (45)  can  be  written, 


±       cos  0  +  sin       ±  j 
tf'jcosg  ±  |)cos  *  ±  COB  (I  T 


LOAD  BALANCE  OF  POLYPHASE  SYSTEMS      323 

or,  choosing, 

E'  -  E, 
thus, 

it  is,  by  (38), 

e'  =  a\e\  ±  a2e2 
where 

=  cos  ("  ±  *}  (47) 


a2  =  cos  I- 


and  the  upper  sign  applies  to  the  reactor,  the  lower  to  the  con- 
denser as  compensating  circuit. 
The  current  then  is 


Jcos        -      + 


(48) 


The  compensating  voltage  e'  thus  can  be  produced  by  connect- 
ing a  transformer  of  ratio  a\  into  the  first  phase,  e\t  a  transformer 
of  ratio,  a2,  into  the  second  phase,  e2,  and  connecting  their  second- 
aries in  series  across  a  reactor  or  condenser  of  suitable  reactance. 

The  current,  i',  in  the  compensating  circuit  consumes  a  current, 
Oil',  in  the  first  phase,  e\t  and  a  current,  ati',  in  the  second  phase, 
e2.  As  the  latter  phase  has  no  load,  the  total  current  in  the 
second  phase  is 

•/        T        /«  _  TT\  /  ,        a  _  3  TT\ 

iz  =  a2i    =  I  cos(-  -}-  2)  cos  ( <£  —  -=  +  -r-l 

the  total  current  in  the  first  phase  is 

•     Q       •  |  •/ 

=  7|  COS  (0  —  a)  -f-  COS   (—  ±  2)    COS  (  ^  —  —  ^  -T- 

=  /{cos(0  -  a)  +  0.5  cos  U  +  ^)  +  0.5  cos  (</>  -  c 

=  0.5/1  COS  (0  —  a)  -h  COS  (0  +  ^ 

V  2> 


hence  has  the  same  value  as  i'2,  but  differs  from  it  by  «  or  90°  in 
phase,  thus  has  to  its  voltage,  e\t  the  same  phase  relation  as  ij 


324  ELECTRIC  CIRCUITS 

has  to  its  voltage,  ez.     That  is,  the  system  is  balanced  in  load,  in 
phase  and  in  armature  reaction. 

In  the  unbalanced  single-phase  load,  the  power-factor  is 

a\  =  cos  a 
in  the  balanced  load,  the  power-factor  is 

/Q      .      7T\ 

01  =  COSl2  ±  4) 
thus,  is  materially  reduced  for  a  reactor  as  compensator,  +^; 

7T 

it  is  in  general  increased  for  a  condenser  as  compensator,  —  -.• 

170.  Instead  of  varying  the  phase  angle  of  the  compensating 
voltage,  ef,  with  varying  phase  angle,  a,  of  the  single-phase  load, 
compensation  can  be  produced  by  compensating  voltages  of 
constant-phase  angle,  utilizing  two  such  voltages  and  varying  the 
proportions  of  their  reactive  currents,  with  changes  of  a. 

Thus,  if 

ii  =  /  cos  (<£  —  «), 
is  the  load  on  phase, 

ei  =  E  cos  </>, 
and  the  second  phase 

e2  =  E  cos       - 


is  not  loaded,  thus  giving  the  unbalanced  power, 

El 

p  =  —  {  cos  a  +  cos  (2  0  -  a)  }  (49) 

2 

as  compensating  voltage  may  be  used, 

the  voltage  of  both  phases  connected  in  series, 

e  =  ei  +  ez 

(50) 


and  the  voltage  of  the  second  phase, 

62  =  Ecos(0-^)-  (51) 

Let,  then, 


be  the  reactive  current  of  the  compensating  phase,  e,  and 
i'z  =  /'2  cos  (0  -  TT), 


LOAD  BALANCE  OF  POLYPHASE  SYSTEMS      325 


=  —  7'2  cos  0 

the  reactive  current  of  the  compensating  phase,  e2. 
The  powers  of  the  two  compensating  circuits  then  are 


p'  =  ei1 


cos  (2  <f>  —  TT) 


cos  2 


and 


p' 


EI' 


cos 


(52) 


(53) 


and  the  condition  of  compensation  thus  is 

7?T  FTf'\./'2  FTf  I  ir\ 

^  cos  (20-  a)  =  -^^cos20  +  -^-2cos(20  -  -)  (54) 
2  Z  &  \  £1 

or,  resolved, 

(7  cos  a  -  I' \/2)  cos  2  0  +  (7  sin  a  -  7'2)  sin  2  0  =  0, 
and  as  this  must  be  an  identity,  the  individual  coefficients  must 
vanish,  that  is, 

7  cos  a 

~^*~  ,  (55) 

7'2  =  7  sin  a  =  I  cos  (a  —  2) 

thus,  the  compensating  voltages  and  currents,  which  balance  the 
single-phase  load, 

ei  =  E  cos  0  .     . 

t,  =  I  cos  (0  -  a)  (56) 


are 


e  =  el 


cos         - 


7  cos  a 


3  TT\ 

-  T) 


(57) 


and 


=       cos 

=  —  7  cos  a  —       cos 
=  7  sin  a  cos 


(58) 


326 


ELECTRIC  CIRCUITS 


As  seen,  this  means  loading  the  second  phase  with  a  reactor 
giving  the  same  volt-amperes, 

El    . 


as  the  unbalanced  single-phase  load  (56),  and  thereby  balancing 
the  reactive  component  of  load,  and  then  balancing  the  energy 
component  of  the  load  by  the  compensating  voltage  e\  +  62,  as 
given  by  (46). 

If  the  single-phase  load   is  connected  across  both  phases  of 
the  quarter-phase  machine  in  series, 


e  =  ei  +  e2 
=  E\/2  cos 


(59) 


in  the  same  manner  the  conditions  of  compensation  can  be  de- 
rived, and  give  the  compensating  circuit, 


(60) 


where 


ET  =  EI. 


For  non-inductive  load, 
this  gives 


a  =  0, 


e' 


that  is,  one  of  the  two  phases  is  compensating  phase  for  the  re- 
sultant. 

171.  As  further  instance  may  be  considered  the  balancing  of 
single-phase  load  on  one  phase  of  a  three-phase  system. 
Let 


ei  =  E  cos  <p 

/         4  TT> 

e$  =  E  cos  ( <£ ;r- 

\  o  - 

be  the  three  voltages  between  the  three  lines  and  the  neutral. 


LOAD  BALANCE  OF  POLYPHASE  SYSTEMS      327 
The  voltage  from  line  1  to  line  2,  then,  is 

(62) 

and  if  a  =  lag  of  current  behind  the  voltage,  the  current  produced 
by  voltage,  e,  is 

i  =  I  cos  ( 4>  +  ^  —  a\ , 
thus  the  power, 

7?7A/2  (  /  *  \  1 

(63) 


and  this  is  balanced  by  the  compensating  voltage  and  current,  as 
discussed  before, 


(64) 

it  is 

p'  =  e'i' 


(65) 


2 
thus, 

Po  =  p  +  p' 


cos  a, 


2 

thus  balanced. 

The  balancing  voltage  (64), 

e'  =  #V3coS(*-f-£ 
lags  behind  the  load  voltage,  e  (62),  by 

f  +  i- 

or  by  half  the  lag  angle  of  the  load,  plus  45°. 
If 

7T 


328  ELECTRIC  CIRCUITS 

or  30°  lag,  it  is 

ef  =  EV3cos(</>  -  g)  (66) 

thus  the  compensating  voltage,  e',  is  displaced  in  phase  from  the 
load  voltage,  eiz  (62),  by  60°,  if  the  lag  angle  of  the  load  is  30°,  and 
in  this  case,  the  second  phase  of  the  three-phase  system  thus  can 
be  used  as  compensating  voltage, 

013   =  €i  —  63 


In  the  general  case,  for  any  lag  angle,  a,  the  compensating  vol- 
tage (64)  can  be  produced  by  the  combination  of  the  two-phase 
voltages,  e\  and  63,  as 


similar  as  was  discussed  in  the  quarter-phase  system. 

The  second  phase,  #13,  as  compensating  voltage,  loaded  by  a  re- 

7T 

actor,  balances  the  load  of  phase  angle,  a  =  ^  or  30°.     For  other 

angles  of  lag,  either  another  phase  angle  of  the  balancing  voltage 
is  necessary,  or,  if  using  the  same  balancing  voltage,  the  balance 
is  incomplete. 

Let  thus: 
the  load 


i  =  I  cos  (<£  +  ^  -  a  j  , 

be  balanced  by  reactive  load  on  the  second  phase, 

cos  (f  -  |)  » 


*'  =  7cos((/> ~^, 


it  is: 

power  of  the  load, 

p 


cos 
•J 

balancing  power, 


a  +  cos  (2  0  +  I  -  a)     J 


cos 


LOAD  BALANCE  OF  POLYPHASE  SYSTEMS      329 


thus,  total  power, 
pQ  =  p  +  p' 


and 


cos  a  +  cos 


cos  a 


f  2<£-f  !  -  aj-  cos  (2<£  +  |j 
+  sin  (  ^  -  |  j  cos  f  2  0  -  |  +  ^  )    J 


sing  -a) 


cos  a 


is  the  ratio  of  the  remaining  alternating  component  of  power, 
to  the  constant  power,  and  may  be  called  the  coefficient  of 
unbalancing. 


CHAPTER  XVII 
CIRCUITS  WITH  DISTRIBUTED  LEAKAGE 

172.  If  an  uninsulated  electric  circuit  is  immersed  in  a  high- 
resistance  conducting  medium,  such  as  water,  the  current  does 
not  remain  entirely  in  the  "circuit,"  but  more  or  less  leaks 
through  the  surrounding  medium.  The  current,  then,  is  not  the 
same  throughout  the  entire  circuit,  but  varies  from  point  to  point: 
the  currents  at  two  points  of  the  circuit  differ  from  each  other  by 
the  current  which  leaks  from  the  circuit  between  these  two  points. 

Such  circuits  with  distributed  leakage  are  the  rail  return  circuit 
of  electric  railways;  the  lead  armors  of  cables  laid  directly  in  the 
ground;  water  and  gas  pipes,  etc.  With  lead-armored  cables  in 
ducts,  with  railway  return  circuits  where  the  rails  are  supported 
above  the  ground  by  sleepers,  as  in  interurban  roads,  the  leakage 
may  be  localized  at  frequently  recurring  points ;  the  breaks  in  the 
ducts,  the  sleepers  supporting  the  rails,  etc.,  but  even  then  an 
assumption  of  distributed  leakage  probably  best  represents  the 
conditions.  The  same  applies  to  low-voltage  distributing  sys- 
tems, telephone  and  telegraph  lines,  etc. 

The  current  in  the  conductor  with  distributed  leakage  may 
be  the  result  of  a  voltage  impressed  upon  a  circuit  of  which  the 
leaky  conductor  is  a  part,  as  is  the  case  with  the  rail  return  of 
electric  railways,  or  occurs  when  a  cable  conductor  grounds  on 
the  cable  armor,  and  the  current  thereby  returns  over  the  armor; 
or  it  may  be  induced  in  the  leaky  conductor,  as  in  the  lead  armor 
of  a  single-conductor  cable  traversed  by  an  alternating  current; 
or  it  may  enter  the  conductor  as  leakage  current,  as  is  the  case 
in  cable  armors,  gas  and  water  pipes,  etc.,  in  those  cases  where 
they  pick  up  stray  railway  return  currents,  etc. 

When  dealing  with  direct-current  circuits,  the  inductance  and 
the  capacity  of  the  conductor  do  not  come  into  consideration 
except  in  the  transients  of  current  change,  and  in  stationary  con- 
ditions such  a  circuit  thus  is  one  of  distributed  series  resistance 
and  shunted  conductance. 

Inductance  also  is  absent  with  the  current  induced  in  the  cable 
armor  by  an  alternating  current  traversing  the  cable  conductor, 

330 


CIRCUITS  WITH  DISTRIBUTED  LEAKAGE      331 

and  with  all  low-  and  medium-voltage  conductors,  with  the  com- 
mercial frequencies  of  alternating  currents,  the  capacity  effects 
are  so  small  as  to  be  negligible. 

In  high- voltage  conductors,  such  as  transmission  lines,  etc.,  in 
general,  capacity  and  inductance  require  consideration  as  well  as 
resistance  and  shunted  conductance.  This  general  case  is  fully 
discussed  in  "Theory  and  Calculation  of  Transient  Electric  Phe- 
nomena and  Oscillations,"  and  in  "Electric  Discharges,  Waves 
and  Impulses,"  more  particularly  in  the  fourth  section  of  the 
former  book. 

173.  Let,  then,  in  a  conductor  having  uniformly  distributed 
leakage,  or  in  that  conductor  section,  in  which  the  leakage  can 
be  considered  as  approximately  uniformly  distributed, 

r  =  resistance  per  unit  length  of  conductor  (series  resistance), 

g  =  leakage  conductance  per  unit  length  of  conductor  (shunted 

conductance), 
and  assume,  at  first,  that  no  e.m.f.  is  induced  in  this  conductor. 

The  voltage,  de,  consumed  in  any  line  element,  dl,  of  this  con- 
ductor, then  is  that  consumed  by  the  current,  i,  in  the  series 
resistance  of  the  line  element,  rdl,  thus: 

de  =  irdl  (1) 

The  current,  di,  consumed  in  any  line  element,  dl,  that  is,  the 
difference  of  current  between  the  two  ends  of  this  line  element, 
then,  is  the  current  which  leaks  from  the  conductor  in  this  line 
element,  through  the  leakage  conductance,  gdl,  thus: 

di  =  egdl.  (2) 

Differentiating  (2)  and  substituting  into  (1)  gives 

ctt 

ar«  =  v.  (3) 

This  equation  is  integrated  by  (see  "  Engineering  Mat  hematics," 
Chapter  II) 

i  =  Ae-'.  (4) 

Substituting  (4)  into  (3)  gives 

a*Ae-al  =  rgAe~al 
hence, 

a  =  ±\/rg, 
and  thus,  the  current, 

(5) 


332  ELECTRIC  CIRCUITS 

where  AI  and  Az  are  determined  by  the  terminal  conditions,  as 
integration  constants. 

Substituting  (5)  into  (2)  gives  as  the  voltage, 


174.  (a)  If  the  conductor  is  of  infinite  length,  that  is,  of  such 
great  length,  that  the  current  which  reaches  the  end  is  negligible 
compared  with  the  current  entering  the  conductor,  it  is 

i  =  0  for  I  =  oo. 
This  gives 

A,  =  0, 
hence, 

i  =  Ae-^1 


e 


(7) 


That  is: 

A  leaky  conductor  of  infinite  length,  that  is,  of  such  great  length 
that  practically  no  current  penetrates  to  its  end,  of  series  resist- 
ance, r,and  shunted  conductance,  g,  per  unit  length,  has  an  effect- 
ive resistance, 


r0  =  (8) 

It  is  interesting  to  note,  that  a  change  of  r  or  g  changes  the 
effective  resistance,  r*o,  and  thus  the  current  flowing  into  the  con- 
ductor at  constant  impressed  voltage,  or  the  voltage  consumed 
at  constant-current  input,  much  less  than  the  change  of  r  or  g. 

(b)  If  the  conductor  is  open  at  the  end  I  =  IQ,  it  is 

i  =  0  for  I  =  Zo, 
hence,  substituted  into  (5) 

0  =  Ai 
and,  putting 

A  =  Ai 
it  is 

1  =  A{  e+v^  (Z°~°  —  e 


e  = 


(9) 


CIRCUITS  WITH  DISTRIBUTED  LEAKAGE      333 


(c)  If  the  conductor  is  grounded  at  the  end  I  =  1Q,  it  is 

e  =  0   for  I  =  I0j 
hence,  substituted  into  (6), 

0  =  An 
and,  putting 

*\.    ~~*    £\.  j6 

it  is 


(10) 

(d)  If  the  circuit,  at  I  =  lo,  is  closed  by  a  resistance,  R,  it  is 

?-  =  R  for  Z  =  Z0, 
hence,  substituting  (5)  and  (6),  gives 


hence, 


Thus, 


e  =  Jl  A{«- 
175.  Substituting, 


(11) 


r°  = 


(8) 


as  the  "effective  resistance  of  the  leaky  conductor  of  infinite 
length," 


334  ELECTRIC  CIRCUITS 

and 


a  =  Vrg  (12) 

as  the  " attenuation  constant"  of  the  leaky  conductor,  it  is 

•j*    -       A\*-<*1  ~   T°  f-a(2lu-l)} 

—        i  "   "D     i 

7?  '  <13) 

e  =  r,A\t-°<  +  „   ,   r» «--«'.-')} 

it  -p  ro 

These  equations  (13)  can  be  written  in  various  different  forms. 
They  are  interesting  in  showing  in  a  direct-current  circuit  features 
which  usually  are  considered  as  characteristic  of  wave  trans- 
mission, that  is,  of  alternating-current  circuits  with  distributed 
capacity. 

The  first  term  of  equations  (13)  may  be  considered  as  the  out- 
flowing components  of  current  and  voltage  respectively,  the  sec- 
ond terms  as  the  reflected  components,  and  at  the  end  of  the 
circuit  of  distributed  leakage,-  reflection  would  be  considered  as 
occurring  at  the  resistance,  R. 

If  R  >  ro,  the  second  term  is  positive,  that  is,  partial  reflection 
of  current  occurs,  while  the  return  voltage  adds  itself  to  the  in- 
coming voltage.  If  R  —  <x> ,  the  reflection  of  current  is  complete. 

If  R<ro,  the  second  term  is  negative,  that  is,  partial  reflection 
of  voltage  occurs,  while  the  return  current  adds  itself  to  the 
incoming  current.  If  R  —  0,  the  reflection  of  voltage  is  complete. 

If  R  =  ro,  the  second  term  vanishes,  and  equations  (13)  be- 
come those  of  (7),  of  an  infinitely  long  conductor.  That  is: 


A  resistance,  R, equal  to  the  effective  resistance, 7-0  =  *-, of  the 

infinitely  long  conductor  of  distributed  resistance  and  shunted 
conductance,  as  terminal  of  a  finite  conductor  of  this  character 
passes  current  and  voltage  without  reflection.  A  higher  resist- 
ance partially  reflects  the  current  and  increases  the  voltage,  and 
a  lower  resistance  partially  reflects  the  voltage  and  increases  the 
current.  Infinite  resistance  gives  complete  reflection  of  current 
and  doubles  the  voltage,  while  zero  resistance  gives  complete  re- 
flection of  voltage  and  doubles  the  current. 


The  term,r0  =  A/-,  thus  takes  in  direct-current  circuits  the  same 

position  as  the  "surge  impedance"  \ta  or  * /^  in  alternating-cur- 

\  (_/        \  i 


rent  circuits. 


CIRCUITS  WITH  DISTRIBUTED  LEAKAGE      335 

176.  Consider  an  instance:  it  has  been  proposed,  for  the  pur- 
pose of  effectively  grounding  the  overhead  ground  wire  used  for 
protection  of  transmission  lines,  to  run  a  bare  underground  con- 
ductor, a  few  feet  below  the  ground  surface,  and  to  connect  the 
overhead  ground  wire  to  the  underground  wire  at  every  pole. 
Assuming  the  underground  conductor  to  be  a  bare  copper  wire 
having  0.41  cm.  diameter,  the  overhead  ground  wire  a  steel  cable 
equivalent  in  conductivity  to  a  copper  wire  of  0.52  cm.  diameter. 
What  is  the  effective  ground  resistance  of  the  underground  wire 
alone,  what  that  of  the  underground  and  overhead  wire  together? 
Assuming  the  leakage  resistance  of  the  underground  wire  to  be 
3  X  10~3  mhos  per  meter? 

The  resistance  of  the  underground  wire  is  1.3  X  10~3  ohms,  that 
of  the  overhead  ground  wire  is  0.82  X  10~3  ohms  per  meter. 

The  effective  resistance  of  one  underground  wire  then  is 


r  =  1.3  X  10~3;    g  =  3  X  10~3,    hence, 
r0  =  0.66  ohm 

thus,  two  underground  wires  in  multiple,  in  the  two  different  di- 
rections, give  an  effective  ground  resistance  of 

0.33  ohm, 
including  the  overhead  ground  wire,  the  resistance  is 

=  0.5  X  10-3;  g  =  3  X  10~3, 


1.3  X  10~3  T  0.82  X  10-3 
hence, 

r0  =  0.41  ohm, 

thus,  the  two  underground  and  two  overhead  wires  together  give 
an  effective  resistance  of 

0.205  ohm. 

This  is  a  very  much  lower  ground  resistance  than  most  local 
grounds  possess. 

Assuming  that  io  is  the  current  which  enters  this  ground  wire 
at  one  point,  I  =  0,  then  the  equation  of  current  distribution,  by 
(7),  is 

r  =  0.25  X  10~3;  g  =  6  X  10~3  (two  in  multiple,  in 

the  two  opposite  directions) 


336 


ELECTRIC  CIRCUITS 


hence 


thus, 


ro  =  JK.  =  0.205 
V0 

a  =  \/rg  =  1.225  X  10~3 


= 


e  =  0.205  i06-l-«M 

where  I  is  given  in  kilometers. 

At  various  distances  from  the  starting  point,  the  current  in 
the  conductors  thus  is: 


distance:  0.  . 
»  X  to  X  l.  . 

0.5 
0.54 

1.0 
0.294 

1.5 
0.155 

2.0 
0.086 

2.5 
0.046 

3.0 
0.025 

4.0 
0.0074 

5  km. 
0.0021 

As  seen,  beyond  2  km.  distance,  the  current  in  the  conductor  is 
practically  nothing. 

177.  If  the  current,  i,  is  an  alternating  current,  and  the  con- 
dition such  that  inductance  and  capacity  are  negligible,  the 
equations  (7),  (9),  (10),  (11)  and  (13)  remain  the  same,  except  that 
ij  e  and  A  are  vector  quantities,  or  general  numbers :  /,  E,  A . 

Considering  thus  the  more  general  case,  where  a  voltage  is  in- 
duced in  the  leaky  conductor.     Such  for  instance  is  the  case  in  the 
lead  armor  of  a  single-conductor  alternating-current  cable. 
Let,  then, 

r     =  resistance  per  unit  length, 

g     =  shunted  conductance  per  unit  length, 

$o  =  voltage  induced  in  the  conductor,  per  unit  length. 


It  is,  then,  in  a  line  element,  dl, 


(14) 


Differentiating  (15)  and  substituting  into  (14)  gives 

E 


0\ 

-  T) 


This  is  integrated  by 


/  -  f  - 


(15) 

(16) 
(17) 


CIRCUITS  WITH  DISTRIBUTED  LEAKAGE      337 

and  by  substituting  (17)  into  (16),  we  get 

a2  =  rg  (18) 

hence,  the  current, 

I  =  Al€-°<  +  42e+<"+|-0  (19) 

where 


and  A  i  and  A2  are  complex  imaginary  integration  constants. 
Substituting  (18)  into  (15)  gives  the  voltage, 

q  =  ro{  A  !€-«<-  A2C+<")  (20) 

where 

ro  -  J-  (21) 


178.  Suppose  now  no  voltage  is  impressed  upon  the  conductor, 
but  the  only  existing  voltage  is  that  induced  in  the  conductor, 
as  for  instance  the  cable  armor. 

(a)  Suppose  the  conductor  is  open  at  both  ends:  I  =  +Zo  and 
I  =  —  Z0,  having  the  length  2  1Q. 

It  then  is 


Substituting  this  in  (19)  gives 

A  !€-<"'  +  A2e+°<°  +  —  °  =  0 


lo   I         =  0 
hence, 


and 

(22) 


in  the  center  of  the  conductor,  for  I  =  0,  it  is 


J 


+ 


1 
\ 


22 


338  ELECTRIC  CIRCUITS 

at  the  ends  of  the  conductor,  f or  I  —  ±  Z0,  it  is 

f-o 


-al0    _|_  €-al( 

hence,  if  the  conductor  is  long,  so  that  e~al°  is  negligible  compared 
with  t+al°,  it  is 

~~r  '         'V^g 
For  an  infinitely  long  conductor,  1Q  =  °° ,  equations  (22)  become 


=  0 


(23) 


as  was  to  be  expected. 

(6)  Suppose  the  conductor  is  grounded  at  one  end,  I  =  0,  and 
open  at  the  other  end,  I  =  IQ.  It  is,  then, 

#  =  o  for  I  =  0 
/  =  o  for  I  =  Zo, 

hence,  the  equations  are  the  same  as  (22).  That  is,  a  conductor 
grounded  at  one  end  and  open  at  the  other  is  the  same  as  a  con- 
ductor of  twice  the  length,  open  at  both  ends. 

A  conductor  grounded  at  both  ends  gives  the  same  equation  as 
an  infinitely  long  conductor  (23). 

Suppose  alQ  is  large,  so  that  e~  oto  is  negligible  compared  with 
e+alof  in  equation  (22).  Then  for  all  values  of  I,  except  those  very 
close  to  IQ,  f]  and  the  exponential  term  of  /  are  negligible. 

That  is,  for  the  entire  length  of  the  leaky  conductor,  except 
very  close  to  the  ends,  it  is,  approximately, 

/  =  £o  I 

'    r  (24) 

$  =  0 

Near  the  ends  of  the  conductor,  where  I  is  near  to  lo,  e~al  is 
negligible  compared  with  t+al,  and  equations  (22)  thus  assume  the 
form, 


TO 
r 


(25) 


CIRCUITS  WITH  DISTRIBUTED  LEAKAGE      339 

179.  As  an  instance,  consider  the  lead  armor  of  a  single-conductor 
cable,  10  km.  long,  carrying  an  alternating  current  such  that  it 
induces  60  volts  per  kilometer.  The  armor  is  open  at  either  end, 
and  of  internal  diameter  of  4.2  cm.,  external  diameter  of  4.6  cm. 
The  leakage  conductance  from  the  cable  armor  to  ground  is  1 
mho  per  kilometer.  What  is  the  voltage  and  current  distribution 
in  the  cable?  What  is  it  with  10  mhos,  what  with  0.1  mho  per 
kilometer  leakage  conductance? 

A  lead  section  of  the  armor  of  (2.32  —  2.  12)  TT  =  2.7  cm.2,  at 
the  specific  resistance  of  lead  p  =  19  X  10~6,  gives:  r  =  0.7  ohm 
per  kilometer. 
It  is,  then, 

r  =  0.7 

Q  =      1 
lo  =     5 
E0  =     60 
thus, 

a  =  0.84 
r0  =  0.84 
hence, 

7  =  86f  1  -  0.015  (e+°-84<  +  €-°-84')) 
#  =  1.08{e 

Thus  the  maximum  current  in  the  cable  armor  is,  I  =  0:7  = 
83.4  amp.,  and  this  current  decreases  very  slowly,  and  is  still,  for 
I  =  2:1  =  79  amp. 

The  maximum  voltage  between  cable  armor  and  ground  is,  for 
I  =  ±5:  E  =  72  volts,  and  decreases  fairly  rapidly,  being,  for 
I  =  ±±:E  =  31.1  volts. 

If  the  cable  is  laid  in  very  well-conducting  soil, 

0-10 

it  is 

a  =  2.65 
r0  =  0.265. 

7  =  86{1  -  1.75  X  10-6(e+2'65'  + 
V  =  40  X  10-6  {e+2'65<  -  e-2'65'  } 

in  this  case,  the  current  is  practically  constant,  I  =  86,  and  the 
voltage  zero  over  the  entire  cable  armor  except  very  near  the  ends, 
where  it  rises  to  E  =  22.7  volts  for  I  =  5.  Within  1  km.  from 
the  ends,  or  for  I  =  4,  it  is  still:  /  =  80;  E  =  1.6.  That  is,  over 


+°-84<        -°-84' 


340 


ELECTRIC  CIRCUITS 


most  of  the  length,  the  cable  armor  already  acts  as  an  infinitely 
long  conductor. 

Hence,  for  values  of  I  near  the  end  of  the  conductor,  /  and  E  are 
more  conveniently  expressed  by  the  equation  (25), 


=    86{1    - 

=  22.7  e- 


(28) 


FIG.  128. 

Inversely,  if  the  cable  is  laid  in  ducts,  which  are  fairly  dry,  and 
the  leakage  conductance  thus  is  only 


it  is 


g    =  0.1 

a   =  0.265 
r0  =  2.65 


CIRCUITS  WITH  DISTRIBUTED  LEAKAGE      341 

hence, 

7  =  86  {  1  -  0.25  (e+°-265<  +  e-°-265<) }    1 

$   =   57    {   c+0.265/   _    C-0.265<J  j 

In  this  case,  the  maximum  voltage  between  cable  armor  and 
ground  is,  at  I  =  ±  5  :  E  =  200. 

As  illustrations  are  shown,  in  Fig.  128,  with  I  as  abscissae,  the 
curves  of  7  and  E,  calculated  from  equations  (26),  (28)  and  (29). 

180.  Considering  now  the  case  of  a  conductor,  which  is  not 
connected  to  a  source  of  voltage,  nor  has  any  voltage  induced  in 
it,  but  is  laid  in  a  ground  in  which  a  potential  difference  exists, 
due  to  stray  currents  passing  through  the  ground.  Such,  for 
instance,  may  be  a  water  pipe  laid  in  the  ground  parallel  with  a 
poorly  bonded  railway  circuit. 

Assuming  the  potential  difference,  eo,  exists  in  the  ground,  per 
unit  length  of  conductor.  The  conditions  obviously  are  the  same, 
as  if  the  ground  were  at  constant  potential,  and  the  potential 
difference,  -e0,  existed  in  the  conductor  per  unit  length.  Thus 
we  get  the  same  equations  as  (22)  and  (23).  If  the  potential 
difference  is  continuous,  as  when  due  to  a  direct-current  railway 
circuit,  obviously  the  quantities  7,  E,  AI  and  A2  are  not  alternat- 
ing vector  quantities,  but  scalar  numbers:  i,  e,  etc.  That  is, 


+  al.    +    €-a/0 
+al    _.    ,-al 


\S        

r 

Assuming  thus  as  an  instance  a  water  pipe  of  5  km.  length: 
IQ  =  5,  extending  through  a  territory  having  50  volts  potential 
difference,  or  :  e0  =  10.  Assuming  that  it  is  connected  with  the 
return  circuit  so  that  there  is  no  potential  difference  at  one  end : 

e  =  0   for  I  =  0. 

Let  the  resistance  of  the  water  pipe  be  r  =  0.01  ohm  per  kilo- 
meter, and  the  leakage  conductance  be  g  =  10  mhos  per  kilometer. 
It  is,  then, 

r    =  0.01 

g  =  10 

lo  =5 
e0  =  10 
thus, 

a  =  0.316 
r0  =  0.0316 


342 
hence, 


ELECTRIC  CIRCUITS 


i  =  1000  {  1  -  0.2  (e+o.«i«  +  e-o.»i«)j 
e  =  6.2  {  e+-81w  -  e-°-8l6M 


hence,  the  maximum  current,  for  I  =  0  :  i  =  600  amp. 
the  maximum  voltage,  for  I  =  5  :  e  =  28.8  volts. 


(31) 


e: 
30 

- 

28 

/ 

26 

i'- 

/ 

24 

600 

~~^ 

'"X. 

/ 

22 

550 

\ 

/ 

20 

500 

t\ 

v 

7 

18 

450 

\ 

/ 

16 

400 

y 

14 

350 

/ 

'  S 

\ 

12 

300 

'/I 

/ 

\ 

/ 

10 

250 

/ 

Y 

/ 

8 

200 

/ 

di 
rfl 

x 

x 

\ 

6 

150 

/ 

-x* 

/ 

\ 

\ 

4 

100 

/ 

^* 

X 

\ 

2 

50 

^ 

£j 

1 

5  1 

2 

6  S 

3 

5  - 

L   4 

•  \ 

0 

0 

FIG.  129. 

As  seen,  a  very  considerable  current  may  flow  under  these 
conditions. 

Fig.  129  shows,  with  I  as  abscissae,  the  current,  i,  and  voltage,  e, 

and  the  current  which  enters  the  conductor  per  unit  length,  -y . 


CHAPTER  XVIII 
OSCILLATING  CURRENTS 

Introduction 

181.  An  electric  current  varying  periodically  between  constant 
maximum  and  minimum  values — that  is,  in  equal  time  intervals 
repeating  the  same  values — is  called  an  alternating  current  if  the 
arithmetic  mean  value  equals  zero;  and  is  called  a  pulsating  cur- 
rent if  the  arithmetic  mean  value  differs  from  zero. 

Assuming  the  wave  as  a  sine  curve,  or  replacing  it  by  the 
equivalent  sine  wave,  the  alternating  current  is  characterized  by 
the  period  or  the  time  of  one  complete  cyclic  change,  and  the 
amplitude  or  the  maximum  value  of  the  current.  Period  and 
amplitude  are  constant  in  the  alternating  current. 

A  very  important  class  are  the  currents  of  constant  period, 
but  geometrically  varying  amplitude;  that  is,  currents  in  which 
the  amplitude  of  each  following  wave  bears  to  that  of  the  pre- 
ceding wave  a  constant  ratio.  Such  currents  consist  of  a  series 
of  waves  of  constant  length,  decreasing  in  amplitude,  that  is,  in 
strength,  in  constant  proportion.  They  are  called  oscillating 
currents  in  analogy  with  mechanical  oscillations — for  instance 
of  the  pendulum — in  which  the  amplitude  of  the  vibration  de-. 
creases  in  constant  proportion. 

Since  the  amplitude  of  the  oscillating  current  varies,  constantly 
decreasing,  the  oscillating  current  differs  from  the  alternating 
current  in  so  far  that  it  starts  at  a  definite  time  and  gradually 
dies  out,  reaching  zero  value  theoretically  at  infinite  time,  prac- 
tically in  a  very  short  time,  short  usually  even  in  comparison 
with  the  time  of  one  alternating  half-wave.  Characteristic  con- 
stants of  the  oscillating  current  are  the  period,  T,  or  frequency, 

/  =  7p,  the  first  amplitude  and  the  ratio  of  any  two  successive 

amplitudes,  the  latter  being  called  the  decrement  of  the  wave. 
The  oscillating  current  will  thus  be  represented  by  the  product 
of  a  periodic  function,  and  a  function  decreasing  in  geometric 
proportion  with  the  time.  The  latter  is  the  exponential 
function,  Af~at. 

343 


344 


ELECTRIC  CIRCUITS 


182.  Thus,  the  general  expression  of  the  oscillating  current  is 

7  =  A'-"  cos  (2irft  -  6). 
Since  A'-«  =  A*  AT*  =  i<rbt, 

where  e  =  basis  of  natural   logarithms,  the  current   may  be 
expressed, 

/  =  i<rbt  cos  (Zwft  -  0}  =  i€~a<t>  cos  (  0-  0), 


X 


360 


540 


Oscillating  E.M.F. 

— 1435 ^ 

E  =  5e         cos0 

-5  dec   8.2° 


w-- 


080 


FIG.  130. 


FIG.  131. 


where  <f>  =  2vft't  that  is,  the  period  is  represented  by  a  complete 
revolution. 

In  the  same  way  an  oscillating  e.m.f.  will  be  represented  by 

E  =  ee-°*cos  (0-  0). 


OSCILLATING  CURRENTS 


345 


Such  an  oscillating  e.m.f.  for  the  values, 

e  =  5,  a  =  0.1435  or  e~2ira  =  0.4,  0  =  0, 

is  represented  in  rectangular  coordinates  in  Fig.  130,  and  in  polar 
coordinates  in  Fig.  131.  As  seen  from  Fig.  130  the  oscillating 
wave  in  rectangular  coordinates  is  tangent  to  the  two  exponential 
curves, 

y  =  ±ee-* 

In  polar  coordinates,  the  oscillating  wave  is  represented  in  Fig. 
131  by  a  spiral  curve  passing  the  zero  point  twice  per  period,  and 
tangent  to  the  exponential  spiral, 


The  latter  are  called  the  envelopes  of  a  system  of  oscillating 
waves.  One  of  them  is  shown  separately,  with  the  same  con- 
stants as  Figs.  130  and  131,  in  Fig.  132.  Its  characteristic  feature 
is:  The  angle  which  any  concentric  circle  makes  with  the  curve, 

y  =  ee~a*,  is 

tan  a  = 


FIG.  132. 


FIG.  133. 


which  is,  therefore,  constant;  or,  in  other  words:  "The  envelope 
of  the  oscillating  current  is  the  exponential  spiral,  which  is  char- 
acterized by  a  constant  angle  of  intersection  with  all  concentric 
circles  or  all  radii  vectores."  The  oscillating  current  wave  is 
the  product  of  the  sine  wave  and  the  exponential  or  loxodromic 
spiral. 

183.  In  Fig.  133  let  y  =  ee~a<i>  represent  the  exponential  spiral; 
let  z  =  e  cos  (0  —  0) 

represent  the  sine  wave; 
and  let  E  =  e<~a*  cos  (0  -  e) 


346  ELECTRIC  CIRCUITS 

represent  the  oscillating  wave. 
We  have  then 

dE 


-  sin  (<f>  —  0)  —  q  cos  (<ft  —  0) 

cos  O  -  0) 

=  -  {tan  (0  -  0)  +  a}; 

that  is,  while  the  slope  of  the  sine  wave,  z  =  e  cos  (0  —  0),  is 
represented  by 

tan  7  =  —  tan  (<f>  —  0), 
the  slope  of  the  exponential  spiral,  y  =  ee~a<t>,  is 

tan  a  —  —  a  =  constant, 
that  of  the  oscillating  wave,  E  =  e€~a<f>  cos  (<£  —  0),  is 

tan/3  =  -  (tan  (<f>  -  0)  +  a}. 

Hence,  it  is  increased  over  that  of  the  alternating  sine  wave 
by  the  constant,  a. 

The  ratio  of  the  amplitudes  of  two  consequent  periods  is 


A  is  called  the  numerical  decrement  of  the  oscillating  wave, 
a  the  exponential  decrement  of  the  oscillating  wave,  a  the  angu- 
lar decrement  of  the  oscillating  wave.  The  oscillating  wave  can 
be  represented  by  the  equation, 

E  =  e€-*tanfltcos(0  -  0). 

In  the  example  represented  by  Figs.  130  and  131,  we  have 
A  =  0.4,  a  =  0.1435,  a  =  8.2°. 

Impedance  and  Admittance 

184.  In  complex  imaginary  quantities,  the  alternating  wave, 

z  =  e  cos  (0  —  0)^ 
is  represented  by  the  symbol, 

E  =  e(cos  0  —  j  sin  0)  =  e\  —  je^. 

By  an  extension  of  the  meaning  of  this  symbolic  expression, 
the  oscillating  wave,  E  =  ee~a<t>  cos  (0  —  0),  can  be  expressed  by 
the  symbol, 

E  =  e(cos  0  —  j  sin  0)  dec  a  =  (e\  —  je2)  dec  a, 
where  a  =  tan  a  is  the  exponential  decrement,  a  the  angular 
decrement,  c~2ira  the  numerical  decrement. 


OSCILLATING  CURRENTS  347 

Inductance 

185.  Let   r  =  resistance,   L  =  inductance,    and    x  =  2wfL  = 
reactance, 

in  a  circuit  excited  by  the  oscillating  current, 

I  =  ie~a*  cos  (0  —  0)  =  z'(cos  6  +j  sin  0)  dec  a 

=  (ii  +  712)  dec  a, 
where      i\  =  i  cos  6,  z'2  =  i  sin  6,  a  =  tan  a. 

We  have  then, 

the  e.m.f.  consumed  by  the  resistance,  r,  of  the  circuit, 

Er  =  rl  dec  a. 

The  e.m.f.  consumed  due  to  the  inductance,  L,  of  the  circuit, 

T  dl  ,T  dl          dl 

Ex  =  L  -j-  =  2  TT/L  -j-  =  z  -r:- 
CM  a</>          a0 

Hence  Ex  =  —  zte-°*{sin  (<£  —  0)  -f  a  cos  (0  —  0)} 

xi>€ — a^ 

= '• sin  (0  —  0  -f  a). 

COS  o: 

Thus,  in  symbolic  expression, 

#x  = |  —  sin  (6  —  a)  —  j  cos  (e  —  a)}  dec  a 

cos  ct 

=  —  xi(a  —  j)  (cos  0  —  j  sin  8)  dec  «; 
that  is,    #*  =  —  xl  (a  —  j)  dec  a. 

Hence    the  apparent  reactance  of  the  oscillating-current  cir- 
cuit is,  in  symbolic  expression, 

X  =  x(a  —  j)  dec  a. 

Hence  it  contains  a  power  component,  ax,  and  the  impedance 
is 

Z=(r  —  X)  dec  a=  {r  —  x(a—  j)}  dec  a  =  (r  —  ax  -}- jx)  dec  a. 

Capacity 

186.  Let  r  =  resistance,  C  =  capacity,  and  xc  =  ~    fn  —  con- 

densive  reactance.     In  a  circuit  excited  by  the  oscillating  current, 
7,  the  e.m.f.  consumed  due  to  the  capacity,  C,  is 


348  ELECTRIC  CIRCUITS 

or,  by  substitution, 

EXc  =  x  I  ie-a<t>  cos  (4>  —  0)  d$ 

=  1    f    .&-"*  {sin  (<f>  -  «)  -  a  cos  (0  -  0)} 


N 
—  0  —  a) ', 


(1  +  a2)  cos  a  " 
hence,  in  symbolic  expression, 

•F*c  =  71 — i — ^ i  ~~  sin  (0  +  «)  —  j  cos   (0  +  a) }  dec  a 

(I  -\-  a*)  cos  a 

=  r— -- — ^  (a  —  j)  (cos  0  —  j  sin  0)  dec  a; 
hence, 

*.•••/  -\      T    J 

v*c  =  i    i    a2^~  a  ~  3)  I  dec  a; 

that  is,  the  apparent  capacity  reactance  of  the  oscillating  circuit 
is,  in  symbolic  expression, 

*<=fT^(-a-^)deca- 

187.  We  have  then: 

in  an  oscillating-current  circuit  of  resistance,  r,  inductive  re- 
actance, x,  and  condensive  reactance,  xcj  with  an  exponential 
decrement  a,  the  apparent  impedance,  in  symbolic  expression,  is, 

/  «\       §  •J'c          /  *\     I      j 


-    jdec  a 

=  ra 
and,  absolute, 


Admittance 
188.  Let 

7  =  ie~a<t>  cos  (<t>  —  0)  =  current. 

Then  from  the  preceding  discussion,  the  e.m.f.  consumed  by  re- 
sistance, r,  inductive  reactance,  x}  and  condensive  reactance,  xc,  is 

E  =  ie-"*  |  cos  (0  -  0)1  r  -  ax  -      °.      2xc1    -  sin  (</>  -  0) 

f  x*     1 1 

Lx  -  ITPJ I 

=  izae-a*  cos  (0  —  0  -f-  5), 


where 


OSCILLATING  CURRENTS 

Xc 


349 


x  — 


tan  5  = 


1  +  a 


r  —  ax  — 


a 


*•  = 


substituting  0  -j-  5  for  0,  and  e  =  iza  we  have 

E  =  ee-a+  cos  (</>  -  0), 


.   (  cos  5         ,  .    sin  5   .    , 

=  ee-a*        -  cos  (<f>  —  0)  +  -    -  sin  (<£  — 

I      Za  2a 

hence  in  complex  quantities, 

E  =  e(cos  6  —  j  sin  6)  dec  a, 


or,  substituting, 

/-•* 


T       „  f  cos  5        .  sin  5  ,    , 

I  =  e{- J  —  \  dec  a; 

(       Za  Za 


r  —  ax  — 


1  +  a 


Xc 


dec  a. 


189.  Thus  in  complex  quantities,  for  oscillating  currents,  we 
have:  conductance, 


r  —  ax  — 


xc 


susceptance, 


x  — 


Xc 


b   = 


1+a2 


admittance,  in  absolute  values. 


350  ELECTRIC  CIRCUITS 

in  symbolic  expression, 


Since  the  impedance  is 

z  -  (r  ~  ax  ~  rrr.' 

we  have 

V  .  .  ^a       z,  Xa 

z'  y  =  2~;  °  =  T»  b  =  r*; 

^  &a  "a  &a 

that  is,  the  same  relations  as  in  the  complex  quantities  in  alter- 
nating-current circuits,  except  that  in  the  present  case  all  the 
constants,  ra,  xa,  za,  g,  z,  y,  depend  upon  the  decrement,  a. 

It  is  interesting  to  note  that  with  oscillating  currents,  resist- 
ance as  well  as  conductance  have  a  negative  term  added,  which 
depends  on  the  decrement  a.  Such  a  negative  resistance  repre- 
sents energy  production,  and  its  meaning  in  the  present  case  is, 
that  with  the  decrease  of  the  oscillating  current  and  voltage, 
their  stored  magnetic  and  dielectric  energy  become  available. 

Circuits  of  Zero  Impedance 

190.  In  an  oscillating-current  circuit  of  decrement,  a,  of 
resistance,  r,  inductive  reactance,  x,  and  condensive  reactance,  xet 
the  impedance  was  represented  in  symbolic  expression  by 

Z  =  r.+jx.  =  (r  -  a*  -  j-J-j  *. 
or  numerically  by 


Thus  the  inductive  reactance,  x,  as  well  as  the  condensive 
reactance,  xc,  do  not  represent  wattless  e.m.fs.  as  in  an  alternating- 
current  circuit,  but  introduce  power  components  of  negative  sign, 


that  means,  in  an  oscillating-current  circuit,  the  counter  e.m.fs. 
of  self-induction  is  not  in  quadrature  behind  the  current,  but  lags 
less  than  90°,  or  a  quarter  period,  and  the  charging  current  of  a 
condenser  is  less  than  90°,  or  a  quarter  period,  ahead  of  the  im- 
pressed e.m.f. 


OSCILLATING  CURRENTS  351 

191.  In  consequence  of  the  existence  of  negative  power  com- 
ponents of  reactance  in  an  oscillating-current  circuit,  a  phe- 
nomenon can  exist  which  has  no  analogy  in  an  alternat- 
ing-current circuit;  that  is,  under  certain  conditions  the  total 
impedance  of  the  oscillating-current  circuit  can  equal  zero: 

Z  =  0. 

In  this  case  we  have 

r-«*-ribx'  =  0;x-rr^  =  0' 

substituting  in  this  equation, 

*-  2  «L;  xc  =  ; 


and  expanding,  we  have 

a  = 


2aL 

That  is,  if  in  an  oscillating-current  circuit,  the  decrement, 

1 


and  the  frequency  /  =  r — j ,  the  total  impedance  of  the  circuit 

is  zero;  that  is,  the  oscillating  current,  when  started  once,  will 
continue  without  external  energy  being  impressed  upon  the 
circuit. 

192.  The  physical  meaning  of  this  is:  If  upon  an  electric 
circuit  a  certain  amount  of  energy  is  impressed  and  then  the 
circuit  left  to  itself,  the  current  in  the  circuit  will  become  oscillat- 
ing, and  the  oscillations  assume  the  frequency,  /  =  j — F>  an<^  the 
decrement, 

a  = 


m-i 


That  is,  the  oscillating  currents  are  the  phenomenon  by  which 
an  electric  circuit  of  disturbed  equilibrium  returns  to  equilibrium. 

This  feature  shows  the  origin  of  the  oscillating  currents,  and 
the  means  of  producing  such  currents  by  disturbing  the  equi- 


352  ELECTRIC  CIRCUITS 

librium  of  the  electric  circuit;  for  instance,  by  the  discharge  of  a 
condenser,  by  make-and-break  of  the  circuit,  by  sudden  electro- 
static charge,  as  lightning,  etc.  Obviously,  the  most  important 
oscillating  currents  are  those  in  a  circuit  of  zero  impedance, 
representing  oscillating  discharges  of  the  circuit.  Lightning 
strokes  frequently  belong  to  this  class. 

Oscillating  Discharges 
193.  The  condition  of  an  oscillating  discharge  is  Z  =  0,  that  is, 

1 
a  = 


If  r  =  0,  that  is,  in  a  circuit  without  resistance,  we  have  a  =  0, 

/  =  -  --  ~7='>  that  is,  the  currents  are  alternating  with  no  decre- 
2 


ment,  and  the  frequency  is  that  of  resonance. 
.If     2r,   J  1  <  0,  that  is,  r  >  2\/^,  a  and  /  become  imaginary; 

T  O   ~~   JL  \  O 

that  is,  the  discharge  ceases  to  be  oscillatory.     An  electrical 
discharge  assumes  an  oscillating  nature  only,  if  r  <  2*f|v     In 

the  case  r  —  2  */~  we  have  a  =  oo  ,  /  =  0;  that  is,  the  current 

dies  out  without  oscillation. 

From  the  foregoing  we  have  seen  that  oscillating  discharges 
—  as  for  instance  the  phenomena  taking  place  if  a  condenser 
charged  to  a  given  potential  is  discharged  through  a  given  circuit, 
or  if  lightning  strikes  the  line  circuit  —  are  defined  by  the  equation, 
Z  =  0  dec  a. 

Since 
/  =  (ii  —  jiz)  dec  a,  $r  =  If  dec  a, 

Ex  =  -  xf(a  -  j)  dec  a,      flc  =  j-qf^i  /(-  «  -  i)  dec  «> 
we  have 

r  ~  ax  ~       Xc  =  0> 


hence,  by  substitution, 

$xe  =  xf  (-  a  -  j)  dec  a. 


OSCILLATING  CURRENTS 


353 


The  two  constants,  z'i  and  i2,  of  the  discharge,  are  determined  by 
the  initial  conditions — that  is,  the  e.m.f.  and  the  current  at  the 
time,  t  =  0. 

194.  Let  a  condenser  of  capacity,  C,  be  discharged  through  a 
circuit  of  resistance,  r,  and  inductance,  L.  Let  e  =  e.m.f.  at  the 
condenser  in  the  moment  of  closing  the  circuit — that  is,  at  the 
time  t  =  0  or  <f>  =  0.  At  this  moment  the  current  is  zero — that  is, 

/  =  jit,  ii  =  0. 
Since  $Xc  =  xf  (—  a  —  j)  doc  a  =  e  at  <£  =  0, 

y? 


dec  a, 


we  have     xiz\/  L  +  az  =  e  or  iz  = 
Substituting  this,  we  have, 
/    --jf 


dec  a,  EXc  =  -  —  ——  -  (1  -ja)  dec  a, 


,          2 

the  equations  of  the  oscillating  discharge  of  a  condenser  of  initial 

voltage,  e. 

Since 


'r2C 
r 


-  1 


we  have 


JL     r    rL 

~  2a~  2  Vr2C 
hence,  by  substitution, 

/S, 

=    -  je  *-  dec  a, 

4L 


the  final  equations  of  the  oscillating  discharge,  in  symbolic  ex- 
pression. 

23 


INDEX 


Admittance,    with    oscillating    cur- 
rents, 348 
Air  gap  in  magnetic  circuit  reducing 

wave  distortion,  145 
Alloys,  resistance,  2 
Alternating  component  of  power  of 

general  system,  317 
current  electromagnet,  95 
magnetic  characteristic,  51 
Alternations  by  capacity  inductance 

shunt  to  arc,  187 
Aluminum  cell  as  condenser,  10 
Amorphous  carbon  resistance,  23 
Annealing,  magnetic  effect,  78 
Anode,  6 

Anthracite,  resistance,  23 
Apparatus  economy  of  constant  po- 
tential,    constant     current 
transformation,  281 
of  monocyclic  -square,  276 
of  T  connection,  265 
Arc   as   alternating    current    power 

generator,  187 
characteristics,  34 
condition  of  stability  on   con- 
stant current,  173 
on  constant  voltage,  169 
conduction,  28,  31,  42 
constants,  36 
effective     negative     resistance, 

191 

equations,  35 
as  oscillator,  189 
parallel  operation  on  constant 

current,  175 
shunted  by  capacity,  178,  184 

and  inductance,  184 
by    resistance    on    constant 

current,  172 

singing  and  rasping,  188,  189 
tending  to  unstability,  164 
transient  characteristic,  192 
as  unstable  conductor,  167 


Arcing  ground  on  transmission  lines, 

199 

Area  of  BH  relation,  53 
Armature  flux  of  alternator,  233 
reactance  flux  of  alternator,  232 
reaction  of  alternator,  236 
Attenuation    constant,    leaky    con- 
ductor, 334 

of  synchronous  machine  oscil- 
lation, 213 


B 


Balance  of  quarterphase  system  on 

singlephase  load,  322 
of  singlephase  load,  319 
of  threephase  system  on  single- 
phase  load,  325 
of  unbalanced  power  of  system, 

319 
Bends  in  magnetic  reluctivity  curve, 

49 

Bismuth,  diamagnetism,  77 
Bridged    gap    in    magnetic    circuit, 
wave  distortion,  148 


Cable  armor  as  circuit,  330 

equation  of  induced  current,  336 
Capacity,  1 

and  inductance  shunting  circuit, 

181 

inductance   shunt  to   arc   pro- 
ducing alternations,  187 
with  oscillating  current,  347 
and  reactance  as  wave  screen, 

154 
in  series  regulating  for  constant 

current,  247 
shunt  to  arc,  178,  184 

to  circuit,  178 
Carbon,  resistance,  21 
Cathode,  6 
Cell,  7 


355 


356 


INDEX 


Characteristic,  magnetic,  50 
Chemical  action  in  electrolytic  con- 
duction, 6 

Chromium,  magnetic  properties,  83 
Circuit  with  distributed  leakage,  330 

magnetic,  43 

Closed  magnetic  circuit,  wave  dis- 
tortion, 139 
Cobalt  iron  alloy,  magnetic,  78 

magnetic  properties,  80 
Coefficient  of  hysteresis,  61 
Coherer  action  of  pyroelectric  con- 
ductor, 19 

Compensating  voltage  balancing  un- 
balanced power,  320 
Condenser,  electrostatic,  9 
power  equation,  319 
tending  to  instability,  164.     See 

Capacity. 

Conductance   with   oscillating    cur- 
rents, 349 

Conduction,  electric,  1 
Conductors,    mechanical    magnetic 

forces,  106 
Constant    component    of   power   in 

general  system,  317 
current  arc,  stability  condition, 

172 

constant    potential    transfor- 
mation, 243,  286 
reactance,  134 

transformer  and  regulator,  250 
magnetic,  77,  87,  88 
potential       constant       current 

transformation,  243,  286 
reactance,  133 

term  and  even  harmonics,  158 
voltage  arc,  stability  condition, 

168 

series  operation,  297 
Continuous  conduction,  32 
Corona  conduction,  29,  42 
Creepage,  magnetic,  57 
Critical  points  of  reluctivity  curve, 

46 

Cumulative  oscillation,  cause,  166 
produced  by  arc,  188 
in  transformer,  199 
surge,  166 


Current    wave    distorted    by    mag- 
netism, 126 


Damping     power     in     synchronous 

motor  oscillation,  210 
winding    in    synchronous    ma- 
chines, 211 

Danger  of  higher  harmonics,  121 
Decrement  of  oscillating  wave,  343 
Demagnetization  by  alternating  cur- 
rent, 54 

temperature,  78 

Diffusion  current  of  polarization,  8 
Direct  current  producing  even  har- 
monics, 159 

Discharges,  oscillating,  352 
Discontinuous  conduction,  29 
Displacement  of  field  poles  eliminat- 
ing harmonics,  120 
of  position  in  synchronous  ma- 
chine, 210 

Disruptive  conduction,  29,  42 
Distortion  of  wave  improving  regu- 
lation in  series  circuits,  311 
of  voltage  by  bridged  magnetic 

gap,  148 

in  constant  potential  con- 
stant current  transforma- 
tion, 290 

Distributed  leakage  of  circuit,  330 
winding,  eliminating  harmonics, 

116 

Double    frequency    armature    reac- 
tion, 240 
peaked  wave,  113 


Economy,  apparatus,  281 
Efficiency  of  electromagnet,  99 

of  monocyclic  square,  277 

of  T-connection,  268 
Electrodes,  6 
Electrolytic  cell,  8 

condenser,  9 

conductor,  442 


INDEX 


357 


Electromagnet,  91 

constant  current,  93 

potential,  98 
efficiency,  99 

Electronic  conduction,  28,  40 
Elimination  of  harmonics  by  alter- 
nator design,  116 
Energy  of  hysteresis,  57 

storage  in  constant  potential 
constant  current  transfor- 
mation, 280 

Even  harmonics,  114,  153,  157 
Excessive  very   high   harmonics   in 
distortion  by  magnetic  sat- 
uration, 140 

Exciting  current  of  transformer  de- 
pending on  wave  shape,  137 
Exponent  of  hysteresis,  66 


Face  conductor  in  alternator,  114 
Faraday's  law   of  electrolytic   con- 
duction, 6 

Ferrites,  magnetic,  80 
Ferromagnetic  density,  45 
Field  flux  of  alternator,  232 
Film  cutout  in  series  circuits,  298 
Flat  top  wave,  111 
Flicker  of  lamps  and  wave  shape,  124 
Flux  distribution  of  alternator  field, 

114 

Fluxes,  magnetic  of  alternator,  232 
Forces,  mechanical  magnetic,  91,  107 
Form  factor  of  magnetic  wave  dis- 
tortion, 127 

Fractional  pitch  armature  winding 

eliminating  harmonics,  119 

Frequency  conversion  in  cumulative 

surge,  166 

of  synchronous  machine  oscil- 
lation, 213 

Friction  molecular  magnetic,  56 
Frohlich's  law,  43 


Gap   in   magnetic   circuit   reducing 
wave  distortion,  145 


Gas  pipes  as  circuits,  330 

vapor  and  vacuum  conduction, 

28,  41 

Geissler  tube  conduction,  29,  42 
Gem  filament  incandescent  lamp,  22 
Grounded  leaky  conductor,  333 


H 


Half  turn  windings,  114 

Hardness,   magnetic,    coefficient  of, 

44 

Harmonics,  effect  of,  121 
even,  153,  157 

separation  by  wave  screens,  157 
Heusler  alloys,  magnetic  properties, 

81 

High  harmonics  in  alternator,  120 
excessive  in  wave  distortion  by 

magnetic  saturation,  140 
by  slot  pitch,  120 
temperature  insulators,  26 
Homogeneous    magnetic    materials, 

55 
Hunting  of  synchronous  machines, 

166,  208 
Hysteresis,  56 

loss  and  wave  shape,  112 


Impedance    and    admittance    with 

oscillating  currents,  346 
of  line  in   regulation  of  series 

circuits,  306 
Induced     current    in    leaky     cable 

armor,  336 
Inductance,  1 

and  capacity  shunting  circuit, 

181 

power  equation,  316 
as  wave  screen,  153 
Induction  motor  magnetic  circuits, 

228 

instability,  164,  201 
Inefficiency  of  magnetic  cycle,  60 
Infinitely  long  leaky  conductor,  332 
Instability  by  capacity  shunt,  180 
of  circuits,  165 


358 


INDEX 


Instability  of  induction  motors,  201 
of  pyroelectric  conductor,  16 
of  synchronous  motor,  208 

Instantaneous  power  of  general  sys- 
tem, 317 

Insulators,  23,  42 

as  pyroelectric  conductor,  25 

Iron  cobalt  alloy,  magnetic,  78 
magnetic  properties,  79 
resistance,  4 


Kennelly's  law  of  reluctivity,  44 


Lag  of  damping  power  in  synchron- 
ous machine,  213 
of  synchronizing  force,  212 
Lamp  circuits  in  series,  297 

equivalent  of  line  impedance  in 

series  circuits,  306 
Law  of  hysteresis,  62 
Leakage,  distributed,  of  circuits,  330 
flux  of  alternating  current  trans- 
formers, 217 

reducing  wave  distortion,  145 
Leaky  conductor,  330,  332,  336 
Load  balance  of  polyphase  system, 

314 
character  determining  stability 

in  induction  motor,  205 
Loop  of  hysteresis,  56 
Loss,  percentage,  in  magnetic  cycle, 

60 

Loxodromic  spiral,  345 
Luminescence  in  gas  and  vapor  con- 
duction, 28 

Luminous  streak  conduction  in  pyro- 
electric conductor,  18 


M 


Magnetic     circuits     of     induction 

motor,  228 
elements,  77 
friction,  56 
mechanical  forces,  107 


Magnetism,  43 

tables  and  data,  87,  88 
wave  distortion  by  saturation, 
128 

Magnetite  arc,  36 
hysteresis,  62 
magnetic  properties,  80 
as  pyroelectric  conductor,  14 

Magnetization  curve,  48 

Magnetkies,  magnetic  properties,  80 

Manganese  alloys,   magnetic  prop- 
erties, 81 
steel,  magnetic  properties,  79 

Mechanical  magnetic  forces,  91,  107 

Mercury  arc  characteristic,  39 

Metals,  resistance,  2 

Metallic  carbon,  resistance,  22 
conductors,  142 
induction,  magnetic,  47 
magnetic  density,  45 

Mixtures  as  pyroelectric  conductors, 
21 

Molecular  magnetic  friction,  56 

Monocyclic  square,   261,   273,   283, 
293 

Mutual  inductive  flux  of  alternator 
armature  reaction,  237 

N 

Negative  resistance  of  arc,  effective, 

191 

Neodymium,  magnetism,  77 
Nernst  lamp  conductor,  13,  24 
Nickel,  magnetic  properties,  81 

steel,  magnetic  properties,  79 
Nominal    induced    e.m.f.    of    alter- 
nator, 236 

O 

Oils  as  insulators,  26 
Open  circuited  leaky  conductor,  332 
magnetic    circuit,    wave    shape 

distortion,  145 
Organic  insulators,  24 
Oscillating  approach  to  equilibrium 

condition,  210 
currents,  343 
discharges,  352 


INDEX 


359 


Oscillations    of    arcing    ground    on 

transmission  line,  197 
in  capacity  inductance  shunt  to 

circuit,  181 
cumulative,   produced   by   arc, 

188 

which  becomes  permanent,  165 
resistance  of  arc,  196 
Outflowing    current   in   leaky    con- 
ductor, 334 
Overshooting  of  alternator    current 

at  load  change,  238 
Oxygen,  magnetism,  77 


Pyroelectric  conductor,  10,  42 

classification,  20 

resistance  increase  by  high  fre- 
quency, 19 

tending  to  instability,  164 
Pyroelectrolytes,  10,  18 


Quarterphase   system   balanced   on 
singlephase  load,  322 


Parallel  operation  of  arc  on  constant 

current,  175 
Peak  of  current  wave  by  magnetic 

saturation,  126 
reactance,  134 
voltage  used  in  arc  starting,  152 

by  magnetic  saturation,  128 
Peaked  wave,  111 
Permanent  instability,  165 

magnetism,  43 

Pitch  deficiency  of  winding  eliminat- 
ing harmonics,  120 
Polarization  cell,  8 

voltage,  7 

Polyphase    constant   current  trans- 
formation, 284,  287 
power    equation,     unbalanced, 

317 

systems,  load  balance,  314 
Position     change     of     synchronous 

motor  with  load,  209 
Power  component  of  reactance  with 

oscillating  currents,  347 
equation    of    singlephase    load, 

315 
of      unbalanced      polyphase 

load,  317 
Primary  cell,  7 
Pulsating  currents  and  wave  screens, 

156 

magnetic   flux    and    even  har- 
monics, 159 


Rail  return  circuit,  330 

Railway  return  circuits,  330,  341 

Rasping  arc,  189 

Reactance  depending  on  wave  shape, 

132 

on  induction  apparatus,  216 
inductive,      constant      current 

regulation,  246,  281 
of  line  in  regulation  of  series 

circuit,  306 

with  oscillating  currents,  347 
self   inductive  and  mutual  in- 
ductive, of  alternator  arma- 
ture, 239 

shunt  in  series  circuit,  298 
regulating    series    circuit   by 

saturation,  302 
of  synchronous  machines,  232 
total,  of  transformer,  224 
of    transformer,    measurement, 

227 

and  short-circuit  stress,  100 
as  wave  screen,  153 
Reactive  power  of  system,  total  and 

resultant,  317 
Recovery  of  induction  motor  after 

overload,  204 
Rectification  by  arc,  32 

by  electronic  conduction,  40 
giving  even  harmonics,  159 
Rectifying   voltage   range  of  alter- 
nating arc,  33 

Reflected  current  in  leaky  conductor, 
334 


360 


INDEX 


Reflection  at  end  of  leaky  conductor, 
334 

Regulating  pole  converter  and  wave 
shape,  123 

Regulation  of  series  circuits  by  react- 
ance shunt,  301 

Regulator,  constant  current,  251 

Reluctivity,  43 
curve,  46 

Remanent  magnetism,  43 

Resistance,  1 

effective,    of   leaky    conductor, 

333 

of  line  in  series  circuits,  306 
negative  effective,  of  arc,  191 

Resistivity,  magnitude  of  different 
conductors,  42 

Resonance  of  transformer  with  har- 
monics of  magnetic  bridged 
gap,  151 

Resonant  wave  screens,  157 

Resonating  circuit,  constant  current 
regulation,   256,   261,   282, 
290 
as  wave  screen,  154 

Resultant  flux  of  alternator,  232 

Rising  magnetic  characteristic,  51 


S 


Saturation  coefficient,  magnetic,  44 
magnetic,  77 

equation  of  wave  shape,  137 
shaping  waves,  125 
of    reactance    shunting    series 

circuit,  302 
value,  magnetic,  46 
Screen,  wave-,  153 
Secondary  cell,  8 
Self    inductive    armature    flux    of 

alternator,  234 
Series  operation,   constant  current, 

297 

constant  voltage,  297 
Shape  of  hysteresis  curve,  68 
Short  circuit  stress  in  transformer, 

99 

third    harmonic   in   alternator, 
244 


Shunt    protective    device    in    series 

circuits,  298 

Silicon  as  pyroelectric  conductor,  13 
steel,  hysteresis,  62 

magnetic  properties,  79 
Sine  wave  as  standard,  111 
Singing  arc,  188 
Singlephase   load,   power  equation, 

315 
Spark  conduction,  28 

discharge  producing  oscillations, 

197 
Speed   change   of   induction   motor 

with  load,  209 
instability  of  motor,  202 
Stability    characteristic    of    arc    on 

constant  current,  173 
on  constant  voltage,  169 
condition  of  capacity  shunting 

arc,  184 

shunting  circuit,  178 
of  induction  motor,  201 
of  parallel  operation  of  arc, 

175 

of  synchronous  machine,  215 
curves  of  arc,  36,  168 

of  pyroelectric  conductor,  20 
Stable  magnetic  characteristic,  54 
Storage  battery,  8 
Streak    conduction    of    pyroelectric 

conductor,  18,  42 
Stream  voltage  of  arc,  35 

of  Geissler  tube,  29 
Susceptance    with    oscillating    cur- 
rents, 350 

Symmetrical  wave,  114 
Synchronizing  force  and  power,  210 
Synchronous     reactance     of     alter- 
nator, 236 
machines,  hunting,  208 

reactance,  232 

motor    tending    to    instability, 
164 


T-connection    of    constant    current 
transformation,    256,    261, 
282,  290 
as  wave  screen,  154 


INDEX 


361 


Temperature    coefficient    of    insula- 
tors, 24 

of  electrolytes,  4 
of  pyroelectrics,  10 
of  resistance,  2 
Terminal  drop  of  arc,  35 

of  Geissler  tube,  29 
Third  harmonic  absent  in  balanced 
three-phase  alternator,  242 
present    in    unbalanced    three- 
phase  alternator,  243 
in  three-phase  winding,  118 
Three-phase    system    balanced     on 

three-phase  load,  325 
winding    and    third    harmonic, 

118 

Transformer,  constant  current,  250 
cumulative  oscillation,  199 
short-circuit  stress,  99 
Transient,  165 

arc  characteristic,  192 
polarization  current,  9 
reactance   of  alternator   arma- 
ture, 240 

Triple     harmonic.     See   Third  har- 
monic. 

frequency   harmonic  of   single- 
phase  load,  241 
True  self-inductive  flux  of  alternator 

armature,  237 
Two  frequency  alternator,  116 


IT 


Unidirectional  conduction  of  arc,  32 


Unipolar  induction,  114 

Unstable  electrical  equilibrium,  165 

magnetic  characteristic,  54 
Unsymmetrical  magnetic  cycles,  73 


Vacuum  arc  characteristic,  39 

conduction,  28 
Vapor  conduction,  28 
Voltage,  wave  distortion  by  bridged 

magnetic  gap,  148 
by  magnetic  saturation,  128, 
143 


W 


Water  pipes  as  circuits,  330 
Wave  screens,  153 

separating  different  harmon- 
ics, 157 

pulsating  currents,  156 
distortion  in  constant   current 

transformation,  290 
improving  regulation  in  series 

circuits,  311 
shape    distortion    by   magnetic 

saturation,  137 
shaping  of,  111 

transmission  in  leaky  d-c.  con- 
ductor, 334 


Zero  impedance  circuits  with  oscil- 
lating currents,  350 


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